So taking the fundamental group of a pointed topological space can be thought of as applying a functor, $\pi_1$, that gives you something in the category of groups. Does $\pi_1$ have an adjoint?
If $\pi_1: Top_* \to Groups$ is a left adjoint of something, then it would preserve all colimits, which it does not, since we need some openness conditions for example. If it is a right adjoint of something then it preserves all limits, but it does not preserve, for example, pullbacks.
This is what makes the Seifert-Van Kampen Theorem somewhat magical, calculating in some circumstances, a non abelian invariant.
If you see $\pi_1$ as a functor from the homotopy category of pointed connected CW-complexes to the category of groups, it is a left adjoint (that does not contradict Ronnie Brown's answer). See https://mathoverflow.net/q/109779/24563 or https://mathoverflow.net/a/45361/24563. The right adjoint is the classifying space functor.