It seems that recurrence relations like Hofstadter's Q-series, in which nested calls to the relation occur, i.e. Q(n) = ...Q(...Q(...)...)..., don't appear to have explicit formulas listed. My introductory textbooks only offers solutions to various linear ones.
Are there examples of non-linear recurrence relations with explicit formulas, and are there any proofs of non-existence of explicit formulas for other non-linear recurrence relations, or are they simply "hopeless" to figure out?
yes, for example the Catalan numbers have a non-linear reccurence relation but there is an explicit formula for them
Perhaps it goes without saying, but if you have any linear recurrence relation you can turn it into a non-linear one. For example, $F(n) = F(n-1) + F(n-2)$ is for Fibonacci numbers. Let $a(n) = b ^{F(n)}$. Then, $a(n) = a(n-1)a(n-2)$, but, since $F(n)$ has an explicit formula, so does $a(n)$. Moreover, there are many sequences that have explicit summation formulas that have second degree or higher recurrence relations. There are many examples in the OEIS, such as sequence A000166, the number of derangements.
