non-contractible space with trivial homotopy groups

A nice example is the (open) long line. Every compact subset of it is contained in a bounded interval which is homeomorphic to $[0,1]$, and thus the homotopy groups are trivial (and the space is additionally locally contractible, even locally Euclidean!). However, it is not contractible, essentially because it's "too long" to contract the whole thing with a single interval.

You can find many examples among finite topological spaces. For information on these, you can browse Jonathan Barmak's LNM book in the subject.