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A few (IPOd) likely contenders:
Google - Currently the second highest tech valuation. They seem to be one of the only companies putting serious resources in shaping the future (self driving cars, mining asteroids). I can totally see Google being the company that introduces personal assistant robots in homes.
Amazon - Brick and mortar stores will continue to get outclassed by the online experience. Amazon will be THE way to shop, including groceries and everyday items. They will grow into their crazy high P/E and begin to expand margins.
Apple - I think Apple will fall for a number of reasons, the biggest being the passing of Steve Jobs. Their products are great but they can't continue to grow the way they have the last 5 years (we are already starting to see this). They will continue to make solid products but I see them in the same vain as Microsoft circa '99.
Facebook - I am a long-term believer of Facebook but acknowledge they are the riskiest of the others listed so far. They certainly have the opportunity to be the biggest though. It really depends on two things: 1) how they fend off competitors and 2) how they monetize. Ya, those are two pretty huge things to depend on but like it or not they have the potential to be the biggest company in the world in 10 years.
A few non-IPO companies that have a shot:
Twitter - I'm not really sure about Twitter. Are they a protocol? Are they a media company? Are they an advertising firm? There is something about them though that sticks. I can totally see them as the biggest social network in 10 years. I can also see them as being a huge advertising firm if they can get metrics of how tweets influence customers (e.g. if Lebron James tweets that he is getting McDonalds for dinner, what percentage increase in sales does McDonalds get within an hour of that tweet?). I can see Twitter being a $100b company in 10 years if they can figure out how to understand their influential data.
Square - This company has the potential to monopolize phone payments. I can see our phones being the leading source of payments in the next few years so I think Square has the potential to get huuuuuge.
Dropbox - I'm not too sure about Dropbox. They provide a great experience but I just don't know that they have the potential to be the biggest company in the world. Data storage will not get them there - they have to expand into something more.
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All that being said, I think Google and Amazon will fight for the biggest valuation in the world in 10 years. (Ignoring companies that haven't been started yet that revolutionize an industry, obviously.)
Discloser: Long Google. Short Facebook via puts.
Apple is a behemoth - they may be at the top of their game but they have a looong looong way to fall if they start to. They may be the most valuable just by how large they are today. I think generally they will smartly stay at what they are good at (consumer devices/software).
Amazon is my vote - they are amazing at scale in almost every way. Im not too keen on seeing this happen but I think they are going to slowly eat almost all of retail. On top of it, it seems as if they are successfully trying to replicate their model for retail across other segments (AWS/digital/etc) - Amazon seems to want to be a provider for everything and the scary part is that they are doing it quite well and quickly.
Google I think has some good things ahead but in a lot of ways I think they have stagnated from a business perspective. Its a bit hard to see this perspective in that they deservedly uphold an amazing reputation amongst the tech community. They have great ideas but they are still 100% in the ad business after all these years and many years of trying diversification. They have great side projects that live great lives today but from a business perspective they are always still side projects. I think Google will see some moderate growth but am not too bullish on their version of glass or auto-cars ever being true business drivers for them.
Dropbox, Square - great services - will remain blips on the radar of history compared to the others.
Google has an amazing advantage at matching user intent with relevant results, some of that intent is commercial. Amazon has tied some of its revenue to the general technological progress by powering the infrastructure of technology companies(AWS). Of course, this is just a bonus to their nearly unstoppable success in selling physical and digital products. The only thing that can stop Amazon is a series of huge missteps on its part.
Also, http://www.forbes.com/sites/gurufocus/2013/01/08/why-warren-...
http://news.ycombinator.com/item?id=5034258
in another thread remarks that Google knows something about how the world works that other companies neglect. That is, Google knows how to leverage government regulations to protect its financial strength while it disrupts industry segments that are not regulated in a way to stop Google from moving in.
For perspective, the top ten tech companies by valuation in early 2012:
http://smallbusiness.chron.com/10-highest-valued-tech-compan...
E-commerce should continue to drive huge revenues, their cloud offerings are supporting a lot of emerging companies and technologies (which hopefully continues), and if they continue physical product offerings (semi-transformational things like the Kindle), they're looking good.
as an aside it'll be interesting to see what social sites do long term - perhaps Palantir's focus on brilliant engineers and consistent growth will let it eclipse Facebook/Twitter/etc at some point?
1. Become agnostic to their own software. What others have said about Dropbox not getting acquired by Apple is a good thing is true. iCloud is specific to the Apple ecosystem. A great company would expand for all and I get that Apple doesn't need to and there are advantages in not doing so, but that leaves a lot of things off the table that they could. Expanding to allow some things to be cross platform will be a huge advantage for them if they can ever do it.
2. Get better at web services. Apple suck balls when it comes to running anything web related. Whether that's an ad company, a social network, or just about anything else they've tried. They should acquire a huge company in the space (something like FB) and expand to make this weakness a strength.
If not Apple (because of said reasons above), I vote for GitHub. Just kidding. I like Github but it'll probably be Tesla or SpaceX