intuition on the fundamental group of $S^1$
I am familiar with the proof that the fundamental group of the unit circle $S^1$ is $\mathbb Z$, yet I couldn't develop intuition for why it is true.
For example, why would I fail if I try to find homotopy from a path that circles $S^1$ once to the path that stays at one point all the time? What brakes the continuity of such a homotopy?
Find a tree and a rope.
Fix the rope to the ground.
Walk around the tree with the rope.
Connect the two ends of the rope.
Try to pull all the rope close to you.
Does it work? No, you are stuck with a rope that goes around the tree once.
For me the mathematical, and also intuitive, reason involves the relation of groupoids to groups.
Here the groupoid $\mathsf I$ has two objects $0,1$ and only one non trivial arrow $\iota: 0 \to 1$. But when you identify $0,1$ you can iterate the image of $\iota$.
The appropriate groupoid theory, leaned from Philip Higgins, is developed in the 1968, 1988 and 2006 editions of the book now titled Topology and Groupoids, but that book is to my knowledge the only topology text in English to explain to students this more powerful result.
The concept of "the fundamental group of a space" is a bit of a misconception, the precise notion being that of "the fundamental group of a space with base point". In 1967, I introduced the notion of the fundamental groupoid $\pi_1(X,C)$ of a space $X$ with a set $C$ of base points, in order to obtain a version of the Seifert-van Kampen Theorem which could compute the fundamental group of the circle, and much more. See the discussion at this mathoverflow question. Choosing just one base point seems to imply that all spaces are assumed path connected.
Grothendieck wrote to me in 1983: "To make an analogy, it would be just impossible to work at ease with algebraic varieties, say, if sticking from the outset (as had been customary for a long time) to varieties which are supposed to be connected. Fixing one point, in this respect (which wouldn't have occurred in the context of algebraic geometry) looks still worse, as far as limiting elbow-freedom goes! Also, expressing a pointed $0$-connected homotopy type in terms of a group object mimicking the loop space (which isn't a group object strictly speaking), or conversely, interpreting the group object in terms of a pointed "classifying space", is a very inspiring magic indeed - what makes it so inspiring is that it relates objects which are definitively of a very different nature - let's say, "spaces" and "spaces with group law". The magic shouldn't make us forget though in the end that the objects thus related are of different nature, and cannot be confused without causing serious trouble." [My emphasis]
To add another point, the pushout of groupoids above is a special case of a pushout of the form $$\begin{matrix}C & \xrightarrow{f} & D\\ \downarrow && \downarrow \\ G & \to & f_*(G)\end{matrix}$$ where $C=Ob(G)$ and $f:C \to D$ is a function, with both $C$ and $D$ regarded also as discrete groupoids, so that $C \to G$ is the inclusion. In the case $G=\pi_1(X,C)$ for a space $X$, and with conditions on $C$, $f_*(G)$ is isomorphic to $\pi_1(Y,D)$ where $Y= D \cup_f X$. Thus the use of $\pi_1(X,C)$ allows us to consider identifications among a set of base points, something not conceivable if you have only one base point. The construction of $f_*(G)$ includes the construction of free groups, free products of groups, free groupoids, ..., one construction using words instead of several, see the books Higgins, T&G, but with a different notation.