In addition to @Abdelmalek Abdesselam's and @Johannes Hahn's good points, I can add a few things from my own context:
One very non-negotiable thing is that Eisenstein series, with a significant role in the spectral theory of automorphic forms, are not in the relevant $L^2$ space, but are only (in an appropriate sense) "of moderate growth".
The simplest explanatory analogue is the case of Fourier transforms on the real line, and the very standard apparatus there (as mentioned by Abdelmalek and Johannes). To make one comparison, we can realize that to apply a tempered distribution $u$ to the Fourier inversion integral $f(x)=\int e^{2\pi ixy}\widehat{f}(y)\,dy$ it is not obvious, and, in fact, not necessarily correct, that $u(f)=\int u(e^{2\pi ixy})\widehat{f}(y)\,dy$. Namely, the exponential is not a Schwartz function.
Nevertheless, the (purely imaginary…) exponentials are bounded, smooth, etc. So, for example, compactly supported distributions can be sensibly applied to them (compatibly with everything else), and can move inside the Fourier inversion integral.
But in many circumstances one wants to have somewhat more general (though still fairly docile) distributions, not just compactly supported ones. Here, already just on the real line (not to mention the automorphic contexts…), it is possible to tell some needless lies. The basis of the possible lies is that (on the real line), continuous, compactly-supported functions are not sup-norm dense in the collection of all bounded continuous functions. (It's not even about measurability….) Thus, the translation action of $\mathbb R$ on bounded continuous functions on it (with sup norm) is not continuous. (E.g., $\sin(x^2)$). This is very bad… and cannot be counted a pathology, but, rather, a misunderstanding of the proper topology.
I think the fundamental point is that the sup-norm closure of continuous, compactly-supported functions is continuous functions going to $0$ at infinity. Thus, the translation action of $\mathbb R$ is continuous on such functions, with sup norm.
That is, rather than looking just at bounded continuous functions on $\mathbb R$, we look at the spaces $V_t$ (with $t$ real…) of continuous functions $f$ such that $\lim_{x\to \infty}\lvert x\rvert^t\cdot \lvert f(x)\rvert=0$. The continuous, compactly-supported functions are dense in each, so the translation action of $\mathbb R$ is (mercifully) continuous on each.
So, the most technically useful, and topologically useful, version of "bounded continuous" is $\bigcap_{t>0} V_t$, which describes a slightly larger space as a (projective) limit….
Hilariously, the colimit over $t$ (an ascending union) does produce the same topology as the naïve version (the latter incorrect on each limitand).