Problems for which there is no algorithm with smallest time complexity Are there any problems where there is no smallest time complexity, so there is a big-$\mathcal{O}$ but no big-$\Theta$?

More formally, I am looking for any problem $p$ where


for any algorithm that solves $p$ in time $f(n)$
there provably exists another algorithm that solves $p$ in time $g(n)$
where $g(n) = o(f(n))$.


An example would be a problem for which there exist algorithms of complexity $\mathcal{O}(n^k)$ for all $k &gt; 0$, but for which there is no algorithm of lower complexity, like $\mathcal{O}(1)$ or $\mathcal{O}(\log n)$.
The existence of such problems follows from the Blum speedup theorem. The theorem shows, for example, that there exists a problem such that for any algorithm solving it in time $T(n)$, there is another one solving it in time at most $\log T(n)$.