My answer is in agreement with Grothendieck that topological spaces may be seen as inadequate for many geometric, and in particular, homotopical purposes. Round about 1970, I spent 9 years trying to generalise the fundamental groupoid of a topological space to dimension 2, using a notion of double groupoid to reflect the idea of ``algebraic inverse to subdivision'' and in the hope of proving a 2-dimensional  van Kampen type theorem. In discussion with Philip Higgins in 1974 we agreed that: 

1) Whitehead's theorem on free crossed modules, that $\pi_2(X \cup \{e^2_\lambda\},X,x)$ was a free crossed $\pi_1(X,x)$-module, was an instance of a 2-dimensional universal property in homotopy theory. 

2) If our proposed theories were to be any  good, then Whitehead's theorem should be a corollary. 

However we observed that Whitehead's theorem was about _relative homotopy groups_. So we tried to define a homotopy double groupoid of a _pair of pointed spaces_, mapping a square into $X$ in which the edges go to $A$ and the vertices to the base point, and taking homotopy classes of such maps. This worked like a dream, and we were able to formulate and prove our theorem, published after some delays (and in the teeth of opposition!) in 1978.  

We could then see how to generalise this to filtered spaces, but the proofs needed new ideas, and were  published in 1981;  this and subsequent work has evolved into the book ``Nonabelian algebraic topology'' published last August. 

Contact with Loday who had defined a special kind of $(n+1)$-fold groupoid for an $n$-cube of spaces led to a more powerful van Kampen Theorem, with a totally different type of proof, published jointly in 1987. This allows for calculations of some homotopy $n$-types, and has as a Corollary an $n$-ad connectivity theorem, with a calculation of the critical (nonabelian!) $n$-ad homotopy group, as has been made more explicit by Ellis and Steiner, using the notion of a crossed $n$-cube of groups.  

Thus we could get useful strict homotopy multiple groupoids for kinds of structured spaces, allowing calculations not previously possible. 

In this way, Grothendieck's view is verified that as spaces with some kind of structure arise naturally in geometric situations, there should be advantages if the algebraic methods  take proper cognisance of this structure from the start. That is, one should consider the data which define the space of interest.