How long does it take to get rich making video games

It depends.  You can start your own indie game company.  However, you need to know what you're doing.  It's hit and miss - Minecraft hit.  Thousands of others missed.  Hire someone who can help you share your idea and creation with the right people!
It could take two years.  Start with an idea.  Write it down.  Find a game engine.  For a 3d game, check out CryEngine 3 (I'm using it right now for an indie game, in progress for over a year), UDK, or even Unity 3D.  I would recommend UDK, not because of its features, but because licensing is extremely easy.
CryEngine 3 has awesome features and in my opinion is the best engine available.  If you own a company, sign an NDA and get started on licensing.  Then build the product.
If you're a teenager (15+), start now.  It's best to start early.  Make an OK product and work from there to start making money.  Low cost indie games are so easy to make money from even if they don't sell well.As of now, CE3 doesn't have the 20% Indie royalty so for now, try UDK or Unity (bleh).If you're under 15, just wait.  You can't create a company and cannot find any workers unless you're extremely intelligent and can hide your age from others, which I would never recommend.