Examples of higher order algorithms ($\mathcal{O}(n^4)$ or larger)
Most simple example gives nested cycles: we know that, if we want, for example, find similar elements for 2 arrays, then we used 2 nested loops and therefore we have $O(n^2)$ complexity.
Hence, every 4 nested loops gives complexity $O(n^4)$, $k$ nested loops $O(n^k)$.
Another examples Karmarkar's algorithm, AKS primality test