Can all math operations be reduced to a sufficiently complex algorithm?

It's a fundamental result in theoretical computer science that there exists a universal algorithm: a computable partial function $U$ with the property that for every computable function $f$ there is an $n$ such that $$ \forall x:~ f(x)=U(n,x) $$ (Essentially, $n$ can be chosen as a program that computes the function, encoded as a number in some appropriate way, and $U$ is the function that tells how to execute programs in some programming language).

This gives at least a partial answer to the question in the title.

However, mathematics can also speak about functions that are not computable, and those are not realized by the usual universal algorithm. But if we want the case where $f$ is not computable to be handled, by fairness we should also allow $U$ to be noncomputable.

Exploring exactly how many functions we can cram into a single $U$ gets us into some twisty little foundational byways. However, it turns out that practically every function encountered in "ordinary" mathematics can be specified unambiguously by a logical formula built from addition and multiplication, the constants $0$ and $1$, and quantification over subsets of $\mathbb C^n$ and functions between such subsets. In this case but we can say in standard mathematics that there is a (non-computable) $\bar U$ such that whenever $f$ is a function with such a specification, there is a number $m$ such that $f(x)=\bar U(m,x)$.

[It is not essential exactly which restrictions we put on the kind of formula that can be used to specify $f$, but we get into foundational trouble unless there are some such restrictions -- more technically there should be a predetermined repertoire of sets that the formula is allowed to quantify over. In fact a more restricted repertoire than the one outlined above can be used, at the cost of less natural specifications for many functions.]