Examples of higher order algorithms ($\mathcal{O}(n^4)$ or larger) In most computer science cirriculums, students only get to see algorithms that run in very lower time complexities. For example these generally are

Constant time $\mathcal{O}(1)$: Ex sum of first $n$ numbers
Logarithmic time $\mathcal{O}(\log n)$: Ex binary searching a sorted list
Linear time $\mathcal{O}(n)$: Ex Searching an unsorted list
LogLinear time $\mathcal{O}(n\log n)$: Ex Merge Sort
Quadratic time $\mathcal{O}(n^2)$: Ex Bubble/Insertion/Selection Sort
(Rarely) Cubic time $\mathcal{O}(n^3)$: Ex Gaussian Elimination of a Matrix

However it can be shown that
$$
\mathcal{O}(1)\subset \mathcal{O}(\log n)\subset \ldots \subset \mathcal{O}(n^3)\subset \mathcal{O}(n^4)\subset\mathcal{O}(n^5)\subset\ldots\subset \mathcal{O}(n^k)\subset\ldots
$$
so it would be expected that there would be more well known problems that are in higher order time complexity classes, such as $\mathcal{O}(n^8)$.
What are some examples of algorithms that fall into these classes $\mathcal{O}(n^k)$ where $k\geq 4$?
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