An example would be base pi, like this:

3021pi = 3pi3 + 0 + 2pi + 1

You might be asking yourself: what's the use of that? Wouldn't any useful number have an infinite pi-cimal representation? Well, yes and no. 0 and 1 are still 0 and 1 in any base. And if you're working with circles, spheres, and "spheres" in more than 3 dimensions, base pi is fairly useful. But it's true that most rational numbers would have an infinite number of digits in base pi.

Base sqrt(2) is even more useful, since it becomes base 2 (binary) if you set every other digit to 0. Think about it:

(ai x 2i) + (ai-1 x 2i-1) + ... + (a1 x 21) + (a0 x 20) ...

is really the same as

(ai x sqrt(2)2i) + (ai-1 x sqrt(2)2(i-1)) + ... + (a1 x sqrt(2)2x1) + (a0 x sqrt(2)2x0) ...

and so you have a system that can comfortably be used for rational numbers, by skipping every other digit, and can also be used to express numbers that have more to do with fractional powers of 2.