Twitter's (TWTR) debut was one of the biggest investing stories of 2013. Its shares closed 74% higher on just the first day of trading back in November. And the stock is now up 42 percentage points since its opening-day closing price.

Given Twitter's remarkable success so far, we thought it might be helpful for our readers if we identified some of the up-and-coming great companies of 2014. Below, our top analysts share their thoughts on which companies might become the "Twitter of 2014." Many of their suggestions are not publicly traded at the moment, so you'll need to follow those companies carefully over the course of the year.

Alex Dumortier: Assuming the euphor... err... excitement regarding technology IPOs continues, online storage company Box ought to receive a warm reception from the stock market this year. Although the calendar for an IPO is not yet known, CEO Aaron Levie said last year that a 2014 IPO was likely; last November, Reuters reported that Box had selected Credit Suisse, JPMorgan Chase, and Morgan Stanley to lead the offering in the first half of 2014, with plans to raise $500 million. The most recent private funding round valued the company at $2 billion -- a Twitter-esque valuation if you consider that the company aimed to bring in roughly $100 million in revenue last year and that the CFO told CNBC last July that he expects Box to achieve profitability "in the next few years."

Box is not without some very valuable assets, however, including a 20 million-plus user base. (Not all are paying customers, mind you -- Box operates on a "freemium" model.) The company has observed that individual users who start with the free service lead to corporate sales once they begin sharing documents with co-workers; indeed, Box already serves 97% of Fortune 500 companies and more than 180,000 businesses altogether. Furthermore, in my experience, online storage customers tend to be something of a captive market: Once you're signed up and the service does the trick, it's going to be tough to convince you to switch to a different provider. That's important for a start-up like Box, which is competing with technology behemoths Amazon.com and Google.

Jim Mueller: When I think of solar power, what first pops into my mind is former President Jimmy Carter in a sweater, encouraging Americans to go green. It's taken nearly 40 years, but SolarCity (SCTY.DL) is on the path to fulfilling Carter's desire.

Historically, the problem with solar electricity has been the cost. Only recently have solar panel prices fallen enough for solar to compete with utilities. Still, installing a solar panel system remains expensive, a high hurdle for many. SolarCity solves this problem.

It installs the panels, paying for everything; the customer doesn't pay a dime. Then, via a contract typically lasting 20 years, SolarCity sells to the customer the electricity generated at a lower rate than the local utility's. Think of SolarCity as a utility, but with dispersed power plants. And just like a utility, I expect it to generate tons of cash.

By the end of 2012, it had installed 288 megawatts of panels, 4% of the total solar capacity installed in the U.S. For 2013, it expects to have installed 278 MW, annual growth of 77%, with similar growth for 2014 expected. In four years, it has accumulated 82,000 customers, with a goal of 1 million by 2018. If it succeeds, today's 380% rise since its first day of trading in late 2012 could be just the beginning.

Utilities are pushing back to protect their own business, of course, but I expect SolarCity to survive as it disrupts how we make electricity. Carter would be happy.

Brendan Mathews: GrubHub Seamless is gearing up for an IPO, possibly as soon as late 2014. The company has a smart, easy solution to facilitate online food orders. It's proven very popular, but it still has plenty of room to grow. When the company comes public, likely in the next year or two, it could be a major hit.

GrubHub Seamless provides online ordering capabilities for small restaurants. Via the company's slick website, hungry consumers can find nearby takeout and delivery options, place an order online, and pay via credit card. Consumers love it for the convenience, and restaurants love it because it brings in business and saves on technology costs. Instead of developing its own online ordering systems -- an expensive proposition for a small business -- the restaurant simply outsources that function to GrubHub Seamless and pays a small fee per order.

The company is a leader in the space -- it has more than 2 million users and more than 12,000 restaurants in 40 cities. But it has only really addressed the tip of the iceberg of the delivery and takeout businesses. It has potential for massive growth as it rolls out service to more cities, gains additional users, and signs up new restaurants.

When shares are offered to investors, I expect many will channel their inner Peter Lynch and buy in based on positive customer experience. The IPO is too far away to speculate on pricing or terms, but I'd expect investors to voraciously devour shares.

Maxx Chatsko: If you like Twitter for being an incredible revenue growth machine, then you will certainly appreciate the awesome potential of Intrexon (PGEN -1.14%), which held an IPO in August. The synthetic biology company designs better biological systems -- stem cells, DNA therapeutics, microorganisms for industrial chemical production, biotech crops, biotech fish, and the like -- to create enhanced biotech products in exclusive channel collaborations, or ECCs. Intrexon recognizes revenue from research and development reimbursement, milestone payments, and, eventually, large royalties on successfully developed products.

While it will take time to develop products and build a portfolio, think of Intrexon as the Intel of biotech (albeit a very early version of Intel). You don't have to understand the ins-and-outs of synthetic biology to appreciate the amazing reach and potential of the company's platform; much the same way you don't need to understand semiconductors or software to realize the power of Intel, Microsoft, or Google. Eventually, products that touch a wide range of industries will emerge from the company's ECCs -- with consumers and investors reaping the rewards.

This year will see a healthy number of catalysts. Intrexon subsidiary AquaBounty Technologies is expected to gain regulatory approval for an engineered Atlantic salmon that grows in half the time of conventional aquaculture salmon. Additionally, Intrexon has continued to pile up partnerships with companies, including proven names such as Johnson & Johnson, and should continue to progress in 2014. It also doesn't hurt that the American bioeconomy -- industrial biotech, biotech crops, and biopharmaceuticals -- reached $350 billion in 2012 and is growing at an annual clip of 15%. Intrexon may be expensive today using traditional metrics (and Wall Street may do more harm than good in the short term), but it is well undervalued for its long-term potential.

Jason Moser: One trend that will continue to play out in 2014 is e-commerce. Given the fact that e-commerce only makes up about 6% or so of the overall U.S. retail market, it seems like the trend is just getting started. And while behemoths like Amazon are setting the stage, smaller players like Wayfair.com are growing quickly as well.

Founded in 2002, Wayfair.com sells midmarket home goods and furniture online. The company also operates Joss and Main, as well as AllModern, helping to expand its customer base. Home and furnishing in the U.S. is a $200 billion market. Wayfair.com grabbed about $1 billion in sales in 2013, up from $600 million in 2012. So the way I see it, one of two things is going to happen for Wayfair.com in 2014. First, the company goes public. Easy peasy lemon-squeezy. But I could actually see another scenario play out. Wayfair's biggest potential drawback today is distribution. It simply doesn't have the same capability that Amazon possesses, which can result in longer waits and more expensive shipping for customers. That's one of the reasons Wayfair.com is actually a third-party seller on Amazon today. So it wouldn't surprise me at all to see Amazon acquire Wayfair in 2014 in an all-stock deal.

Wayfair is at a very similar stage of its life as Zappos was in 2009 before Amazon bought the online shoe and clothing retailer. With Amazon shares at all-time highs, the company could take advantage of "cheap" currency.

Justin Loiseau: This year, tourists from around the world are heading to Colorado to get high -- and it has nothing to do with elevation. On Jan. 1, Colorado became the first state to legally allow recreational marijuana use, after overcoming a long legislative battle and steep opposition.

While Colorado residents can only buy 1 ounce at a time from just 30 stores, this legalization could change the long-term game for marijuana markets. In the next year, medical marijuana and hemp stocks like Medical Marijuana and CannaVest are going to get a lot of attention.

Both these corporations are small caps with volatile share prices. Medical Marijuana shares trade 54% higher than a year ago, while CannaVest is up a whopping 470%.

While Colorado's legalization won't change acceptance and uptake overnight, the two companies may experience supply-side gains from increased production and industry-specific technological advancements, while more general exposure may push consumers to demand more marijuana- and hemp-based products in the years to come.

The future of these companies and their shares is highly uncertain, but one thing's for sure -- hemp and marijuana will be high on everyone's discussion list for 2014.

Eric Volkman: Nobody's sure if 2014 will be the year do-it-yourself vacation rental website Airbnb goes public. As usual with companies in the still-developing stage, it's being cagey about when -- and even if -- it'll come to market. I think it's inevitable, and I suspect the IPO will be sooner (read: this year) rather than later.

This is because (a) the company has some heavy venture capital firepower behind it that knows how to time an exit from an investment (Sequoia Capital and Andreessen Horowitz, to name two of several), and (b) it's rapidly approaching the size and scope that makes a big-ticket listing appropriate. Airbnb should be a hot issue, not only because of high name recognition and generally positive consumer perception, but also thanks to a solid business model and some impressive numbers.

As of early 2013, the company had middlemanned accommodation for 4 million guests, an impressive 3 million of which came in 2012. The company, which was founded in 2008, makes its money by taking a cut of both the renter's cost and the landlord's take, which makes for an easy-to-understand and compelling investment case. Projections in Internet-land have it that the company could see a top line of $1.35 billion this year, which sounds optimistic to me. But that growth is strong, and more than a few people I know are happily using the service. It's not only apartment owners and travelers that are going to benefit from Airbnb; look for investors to reap some nice rewards, too.

Patrick Morris: While I haven't taken advantage of the recent IPO frenzy, one company that would pique my interest is mobile payments company Square, which has been rumored to be considering an IPO in 2014.

I make my best effort to frequent small, independently run local restaurants, breweries, and even a few stores, and seemingly all of them have turned to using Square to accept payments. I feel as though there isn't a week that goes by when I'm not at one time or another greeted with its newest Stand product that effectively turns an iPad into a cash register capable of accepting payments.

I know Square is easy for businesses to use -- with its reasonable flat fee per swipe, its next-day deposits, and the fact it accepts all major credit cards -- but it's also easy for consumers. Gone are days of printed receipts and leaving tips with pen and paper; instead, we now can sign with our fingers and tip a certain percent with a tap on the screen. Any business that makes the lives of both its clients and its clients' customers better is always compelling.

Square did draw the ire of some clients when it did away with its singular monthly pricing option, and a few people have questioned the leadership of CEO (and Twitter co-founder) Jack Dorsey, but those are all things to simply be monitored, and don't represent major problems, in my opinion.

In 2012, Square was rumored to be valued at $3.25 billion, so there is no telling what it may be worth now. As with any IPO, its price relative to its intrinsic value will definitely be a key consideration. But when you consider the success of MasterCard -- which is up 17.5 times since it IPO'd in 2006 to a value north of $100 billion -- Square could be attractive even if it is a touch "pricey."

David Hanson: Just like Patrick, I can see Square being a big story in 2014.

Most people know Square from the company's white dongles attached to merchants' iPads or iPhones at the checkout counter. However, contrary to what most people think they know about Square, the company is not a credit card processor and doesn't compete with Visa or MasterCard. Square is an aggregator. For a fee, Square handles payment and makes sure all parties are paid appropriately. The fee isn't a bargain-basement price, but the service is ultraconvenient.

Square has all the makings of a hot IPO: a big-name CEO and consumer appeal (as well as a business that reportedly is still unprofitable). However, in the eyes of this Fool, the company's competitive moat is narrow and margins will be under constant attack from eBay's PayPal and Amazon, which is keen on getting involved in the payments space. Despite Square's one-stop-shop convenience today, ultimately, merchants will move toward the low-cost option and a price-war could ensue. Visa, MasterCard, and the banks are always going to get their fat cut when someone swipes their card, but Square may not be as lucky. Square may go public in 2014 to much fanfare, but I will most likely be watching from the sidelines.

Austin Smith: 2013 was the year of the IPO, but I'm still waiting for my favorite to hit the market: Redfin. The redefined real-estate brokerage is bringing the freshest take I've seen to an industry that in many ways hasn't changed in 50 years.

The renaissance in real estate values the past few years may be the catalyst for bringing this company public. According to the Case-Shiller index, home prices were up 13.6 % in 2013. Shares in housing-related stocks soared as a result. Zillow was up 194.5% in 2013, Trulia was up 117.2%, and Lumber Liquidators popped 94.8%. It's my belief that a continued recovery, coupled with seeing these already impressive returns, will light a fire under Redfin to join the party.

Fortunately, there are a lot of signs that the company is doing just that. Redfin's $50 million financing round in November and the hiring of former Zappos exec Chris Nelson as CFO seem to bode well for an IPO. With revenue growth that's rumored to be north of 50%, a customer-centric culture, a killer product, and direct access to MLS inventory, I'm eagerly awaiting an S-1 from this potential highflier in 2014. 

David Gardner's favorite stock for 2014
Motley Fool Co-founder David Gardner is really excited about one un-hyped stock that could soar higher than Twitter in the years to come. To learn more about David's favorite stock, click here

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