Are there any limits that can be solved using the techniques learned in a calculus one class (so allowing things like L’Hospital’s Rule and the squeeze theorem, but not Taylor series expansion), but which are by no means routine/have an interesting trick to them?
I've seen examples where blindly applying L’Hospital’s rule gives an infinite loop, but those typically can be manipulated into a form where things are easier without issue; I'd like something a little more clever than that, but that a highly motivated calculus student could perhaps handle- the sort of thing that could be used as a fun bonus problem.
I think you may be limiting yourself, as limits where things like l'Hopital's rule break down are interesting/useful limits for students to do. Moreover, if you eliminate things like Taylor Series (which could make sense for a Calculus I student), you have eliminated another class of interesting limits. But there are still many interesting limits out there! But of course, to do an interesting limit may involve either specialized knowledge or giving the student a hint.
If you want to look through lots of such examples - with solutions - I'd recommend looking through many of the interesting limits people come up with on Brilliant.org There you can find interesting limits like....
$$ \lim_{n \to \infty} \left(1+ \sum_{r=1}^n \dfrac{1}{3^r r!} \prod_{i=1}^r (2i-1) \right) $$ $$ \lim_{n \to \infty} \sqrt[n]{2020^{2021n} + 2021^{2020n}} $$ etc. You may also consider looking through old Putnam exams and seeing if there are any interesting limits that are at the level you desire - same with old Olympiad problems. But of course, these take the level of difficulty usually well beyond a typical problem you'd even give as a bonus (depending on your students).