Example of a utility maximization problem with a non-binding budget constraint
Assuming positive prices, any utility function that doesn't satisfy strong monotonicity of at least one good would have a non-binding budget constraint. Since you asked for some examples:


$U(x, y) = 0$,
$U(x, y) = -x$.


Re: Giskard's comment.

I'm not sure I used the correct term here when I said "strong monotonicity of at least one good." To be precise, what I meant to say was:

Let $U(x_1, x_2)$ be the utility function. The function would have non-binding budget constraint if there are some $(x_1, x_2)$ such that $dU/dx_1 \leq 0$ AND $dU/dx_2 \leq 0$.