I'd like to expand on Dustin's point. There is simply no way to think sensibly about equivariant topology, whether algebraic or geometric, without taking account of multiple basepoints. Even taking account of them one runs into subtle difficulties invisible without them (see eg [65] on my web page). I'll give examples from algebraic topology, since that is what I know best, but examples from geometric topology must abound, as illustrated in other answers.

Take a compact Lie group, or even just a finite group, and consider a smooth closed $G$-manifold $M$. What does it mean for $G$ to be orientable, and what is an orientation? These are seriously interesting questions, necessary to make sense of equivariant Poincar'e duality, and they are difficult except in the boringly simple-minded case (treated in [53] on my website) when the tangent $G_x$-representation $T_x$ is isomorphic to the restriction to $G_x$ of an ambient $G$-representation $V$ for all $x\in M$. Usually there is no such $V$, and then I can't imagine answers that do not use functors defined on equivariant fundamental groupoids (which themselves are not altogether obvious to define.) Three references which give rather different answers to these questions are [93] and [100] on my web site, and http://front.math.ucdavis.edu/0310.5237, by Costenoble and Waner. I actually do not know how to compare these answers or to calculate with them.

Again, while one can (twistedly) escape explicit use of fundamental groupoids when setting up the Serre spectral sequence with local coefficients nonequivariantly, one cannot do so equivariantly.

Perhaps invoking equivariant theory is overkill, but the fundamental groupoid is such a natural thing, and so elementary, that it seems a little perverse to try to avoid it!

I'd like to expand on Dustin's point. There is simply no way to think sensibly about equivariant topology, whether algebraic or geometric, without taking account of multiple basepoints. Even taking account of them one runs into subtle difficulties invisible without them (see eg [65] on my web page). I'll give examples from algebraic topology, since that is what I know best, but examples from geometric topology must abound, as illustrated in other answers.

Take a compact Lie group, or even just a finite group, and consider a smooth closed $G$-manifold $M$. What does it mean for $G$ to be orientable, and what is an orientation? These are seriously interesting questions, necessary to make sense of equivariant Poincar'e duality, and they are difficult except in the boringly simple-minded case (treated in [53] on my website) when the tangent $G_x$-representation $T_x$ is isomorphic to the restriction to $G_x$ of an ambient $G$-representation $V$ for all $x\in M$. Usually there is no such $V$, and then I can't imagine answers that do not use functors defined on equivariant fundamental groupoids (which themselves are not altogether obvious to define.) Three references which give rather different answers to these questions are [93] and [100] on my web site, and Equivariant ordinary homology and cohomology, by Costenoble and Waner. I actually do not know how to compare these answers or to calculate with them.

Again, while one can (twistedly) escape explicit use of fundamental groupoids when setting up the Serre spectral sequence with local coefficients nonequivariantly, one cannot do so equivariantly.

Perhaps invoking equivariant theory is overkill, but the fundamental groupoid is such a natural thing, and so elementary, that it seems a little perverse to try to avoid it!