Game AIs, Stock market bots, character recognition, adaptations to game of life, computer art wherein you get an AI to paint or to draw and can seed it values. There's lots of cool stuff you can hack away at. Even training a neural net to recognise a '3' is quite interesting.
These things suck. I want a robot with sensors, the ability to move and an arm that can be programmed with a language that is appropriate for AI. That doesn't sound like something technically difficult.
So, what's holding you up exactly? RC cars are cheap, people have done cool stuff with old android phones for sensor packages. If you need more horsepower, stream the data back to a PC, you've got wifi on the phone. New industrial arms are expensive, but you can scrounge one, or get a hobby one. sparkfun had a uArm that would probably work for you.
Because I am a software developer and intelligence hobbyist (from the biology/ethology camp.) I don't know a damn thing about RC cars or android phones nor do I have the time or desire to learn. Sadly, the hardware tinkerers don't know a damn thing about programming intelligent behavior. Until there is some sort of API to connect hardware to a software environment programmable by specialists in that domain, hobbyist robotics will remain in the realm of Battlebots.
Hmm. There are a few options. You could get a roomba, and put a laptop on top of it, use the laptop camera to sense, and the serial port to command the motors. I kinda think you could use python to glue everything together.
Mindstorms are really easy to get started with if you want more control over what the robot will look like, but you'll be facing more mechanical engineering problems.
diydrones has a lot of good information about getting things connected. Most people look to RC cars because they're so inexpensive.
So, the big thing with hardware is it kind of sucks. It always feels like using a butterknife to tighten screws. Things are challenging in ways you don't expect. With software, if you make a mistake, you fix it and recompile. With hardware, you hope nothing breaks.
I too wish it were easy. It's not, it's complicated in ways that suck. If you have money to throw at the problem, i bet you can find a nice industrial platform that's built like a tank that has a nice API. That kind of stuff is thousands of dollars, and i'm not familiar with it.
Maybe I'm misunderstanding you, but robotics is now, and probably always will be, inherently about combining skills from electronics, mechanics, and software. I'm not sure what you're looking for, a complete robot that you can simply program? You could look into the NAO robot[1], probably too simple for you, but I think they're pretty cool. Wish I could afford one ;)
