You can only avoid the recursion limit in cases where dynamic programming would also work, as you have to explicitly call the function in reverse stack order to avoid having the stack build up. If you want fib(10000) you need to call fib(1) through fib(9999) first, as if you were implementing a dynamic programming solution.
This isn't dismissive. lru_cache is one of my favorites too, but it has limitations.
But that isn't a limitation of lru_cache, for example the same higher order function when used in Clojure i.e. memoize with recur for tail recursion will not cause stack overflow. The stack build up is because python doesn't support tail call optimization, not a limitation of lru_cache, just wanted to make it clear because you can use similar higher order functions in other languages which support tail call optimization without any limitations. Deep recursion in Python without sys.setrecursionlimit() is probably not a good idea, memoization can't help you in that. My point was geared towards presenting this pattern of memoization using a higher order function + recursion as an alternative to dynamic programming and in languages with tco and immutable data structures it works beautifully :)
This isn't dismissive. lru_cache is one of my favorites too, but it has limitations.