I certainly see no difficulty in making a robot that wants 8+ analog inputs. One of our earliest analyses with sensors was to figure out which things we could do using DIO or a camera, and we managed to get down to four analogs (two rangefinders to square up on totes, a tape measure potentiometer for the lift, and a gyroscope).
There are a variety of Arduino solutions, including my favorite, the sparkfun pro micro which can give you four analog inputs in a package not much bigger than a stick of gum, and send it over I2C or USB or probably a DIO line if you were willing to write a driver. Their pro-mini has 8 Analog pins and isn't much bigger, and only costs $10, though it does not have USB. If you need many more than that, the Leonardo and Due support 12 analog inputs, and the Mega has 16 analog inputs. When driving an arduino on an FRC robot, I would advise an external regulator, or at least some sort of voltage divider so you're feeding the tiny on-board regulator between 8 and 10 volts rather than over 12. If you hurry to a closing Radio Shack, you can get one of
these on closeout.
In addition to Arduinos (look at the micros as well as the original), you could use an xBee module, which I believe can have 7 analog inputs. You wouldn't be using the radio on it, just sending the inputs to the RIO. Disclosure: I've never done this, but it seems like it should work.
I'm not familiar with STAMP or the full variety of boards out there, but a bit of googling should turn up a dozen different ways to get there.