What is infinity divided by infinity?

Essentially, you gave the answer yourself: "infinity over infinity" is not defined just because it should be the result of limiting processes of different nature. I.e., since such a definition would be given for the sake of completeness and coherence with the fact "the limiting ratio is the ratio of the limits", your

$$ \frac{1 + 1 + \cdots}{2 + 2 + \cdots} = \lim_{n \to \infty} \frac{n}{2n} = \frac{1}{2} $$

and, say (this is my choice)

$$ \frac{1 + 1 + 1 + \cdots}{1 + 2 + 3 + \cdots} = \lim_{n \to \infty} \frac{n}{n(n+1)/2} = 0 $$

would have to be equal (as they commonly define $\infty/\infty$), which does not happen.

I will quote the following from Prime obsession by John Derbyshire, to answer your question.

Nonmathematical people sometimes ask me, “You know math, huh? Tell me something I’ve always wondered, What is infinity divided by infinity?” I can only reply, “The words you just uttered do not make sense. That was not a mathematical sentence. You spoke of ‘infinity’ as if it were a number. It’s not. You may as well ask, ‘What is truth divided by beauty?’ I have no clue. I only know how to divide numbers. ‘Infinity,’ ‘truth,’ ‘beauty’—those are not numbers.”

To elaborate a bit on the comment by sos440, there are at least two approaches to the issue of infinity/infinity in calculus:

(1) $\frac \infty\infty$ as an indeterminate form. In this approach, one is interested in the asymptotic behavior of the ratio of two expressions, which are both "increasing without bound" as their common parameter "tends" to its limiting values;

(2) in an enriched number system containing both infinite numbers and infinitesimals, such as the hyperreals, one can avoid discussing things like indeterminate forms and tending, and treat the question purely algebraically: for example, if $H$ and $K$ are both infinite numbers, then the ratio $\frac H K$ can be infinitesimal, infinite, or finite appreciable, depending on the relative size of $H$ and $K$.

One advantage of approach (2) is that it allows one to discuss indeterminate forms in concrete fashion and distinguish several cases depending on the nature of numerator and denominator: infinitesimal, infinite, or appreciable finite, before discussing the technical notion of limit which tends to be confusing to beginners.

Note 1 (in response to user Xitcod13): Here an infinitesimal number, in a number system $E$ extending $\mathbb{R}$, is a number smaller than every positive real $r\in\mathbb{R}$. An appreciable number is a number bigger in absolute value than some positive real. A number is finite if it is smaller in absolute value than some positive real.