The first thing that came to my mind (except of a definite integral mentioned in this video) is a limit of a function f(x) at x -> 0 (or x -> infinity, or any other number).

Another one: Taylor series (or MacLaurin series, its particular case). Strictly speaking, this is not a functional, because you get a *function* from a function, not a number. But a value of that polynomial at exact point will be a functional.

At complex analysis we can meet a residue of a function