I think the question needs to be sharpened to exclude algorithms that compute(or even involve)combinatorially complex structures.For example, the convex hull of $n$ points in $\mathbb{R}^d$ has size $\Theta(n^{\lfloor d/2 \rfloor})$for fixed $d$.There is an asymptotically optimal algorithm (due to Chazelle (1)) to compute this hullin time $O( n \log n + n^{\lfloor d/2 \rfloor} )$.So one could exceed any power of $n$ in the time complexity by selection of a sufficientlylarge $d$.
So you need to specify that the algorithm is a decision procedure,outputting only one bit, Yes or No.But even here, there is no upper bound on the "worst" algorithm time complexity.
Again consider the convex hull in $\mathbb{R}^d$, $d$ fixed, and ask:(a) Is the hull simplicial?or (b) Does the hull have exactly $F$ facets?Jeff Erickson showed (2) that, even for these decision questions,$\Omega( n \log n + n^{\lceil d/2 \rceil -1} )$ time is needed,matching the known upper bounds for odd $d$.
(1) Bernard Chazelle."An optimal convex hull algorithm in any fixed dimension."Discrete & Computational Geometry,Volume 10 (1993), Number 4, 377–409.Zbl 0786.68091
(2) Jeff Erickson."New Lower Bounds for Convex Hull Problems in Odd Dimensions."SIAM J. Comput., 28 (1999), 1198–1214.Zbl 0939.68047
An irrelevant aside: I coauthored an algorithm with time complexity $O(n^{42})$. :-)