Interesting examples where $\frac 0 0$ is defined. After giving up some nice algebraic properties we could give examples where $\frac a 0$ is defined for nonzero $a$, for example consider the real line with one point compactification.
Just for curiosity, are there any interesting examples in which we give up the least amount of good algebraic properties and make the expression $\frac 0 0$ defined?(that is, there is a unique element $k$ such that $0\cdot k$ is $0$) By interesting I mean nothing like the trivial ring.
I am thinking I should consider multiplication where $0\cdot x$ is only zero for $x=k$ but I’ve run out of examples.
Some of these are not your exact $\frac 00$ case, but they are still wondrous:
1.Here is wheel theory. This establishes some rules for division by zero.
2.The function $x^x$ has the limit for x=0 as $0^0\to 1$. Graph
3.The function $\frac 1x$ has the limits to 0 from the positive side as $\infty$ and $-\infty$ from the other side. Here $\frac 10$ can be defined by complex infinity
4.Here are some other possibilities and their discussion of this case. It has rules for indeterminate, shown by ¿, complex infinity, shown by an infinity with a tilda over it, and more!
5.Here is a Wikipedia page discussing division by zero here.
6.This case in your title appears in many limits such as these.
7.Best for last, the function $x^{{{\frac 1x}^ {\frac 1x} }^{…}}$ evaluated at x=0 gives:$$0^{{{\frac 10}^ {\frac 10} }^{…}}\to 0$$
Here is the graph of this function

Here is a nice limit. You should see it in the linked answers about a limit. If you apply direct substitution you get $$\frac{1^\frac 10-1^\frac 10}0\ \to \frac {3e^2}4$$

I could go on forever. I will add more. As always, please correct me and give me feedback!
Perhaps dy/dx is an interesting example of what happens when we reach 0/0.
As you will know it is well defined for most differentiable algebraic formula, but with a wide variation in the particular answer you get.
Also try considering if it is identical to (1/0)*(0/1) and variants on that theme.
E.g. If infinity (1/0) is important to study, why aren't infinitesimals just as interesting, etc.