Is there a problem which can be solved in $O(f_k(n))$ time for any $k$ where $\lim_{k \to \infty} f_k(n) = 0$ but no constant time solution is known? So Fibonacci heaps can perform insertions in amortized constant time, and I came up with a sequence of heap data structures $H_k$ such that insertions can be performed in amortized $O((\log n)^{1/k})$ time where $n$ is the size of the heap. If we didn't know insertions could be performed in constant time, then my result would suggest it, since we can get arbitrarily close to constant running time asymptotics with my solution. 

This led to me to wonder, is there an example of a family of algorithms or data structure operations, that are all functionally equivalent, and can be performed in $O(f_k(n))$ time, where $f_k$ is a sequence of functions such that $\lim_{k \to \infty} f_k(n) = 0$ for all $n$, but no constant time alogorithm/data structure operation can be found? Or at least where no constant time solution is currently known? I suspect the answer is yes but probably hard to prove. It would be very surprising if the existence of such family of algorithms with $O(f_k(n))$ running time would imply a constant time algorithm exists.
You can sort numbers in $O(\frac{n\log n}{k})$ for all fixed $k$, and $\lim_{k\to\infty} \frac{n\log n}{k} = 0$ for all fixed $n$.