1. Like other people mentioned, math is not just about numbers, so that won't work for fields of math without addition.
2. If you want to reduce equations dealing with irrational numbers down to additions/subtractions, then you'll have to deal with approximations or infinite decimals/infinite sums which, misleadingly, are not really sums. An infinite sum is usually defined as the limit of the sequence of partial sums and I don't think limits can't be reduced to addition (since you now have logical quantifiers and sequences).
3. If you want to go deeper, addition itself is just a form of repeated "successor". For example, 3 + 2 is just the successor of the successor of 3.
However, this train of thought is very interesting in my opinion and we can follow it while sticking to the natural numbers.
A particular class of functions that I think you would find interesting are the primitive recursive functions which are:
* constant functions (always returns the same thing)
* the successor function (returns the number that comes after)
* projections (returns one of the inputs, for example, f(x, y) = y would be a projection)
* compositions of primitive recursive functions (for example x + (y \* z), multiplication followed by addition).
* functions defined using "primitive" recursion. By "primitive" I mean that the function has to be defined for 0, and then the function for n can only be defined in terms of the function for n-1. For example, multiplication is usually defined using a primitive recursion. x \* 0 = 0 and x \* y = (x \* (y-1)) + x). Times 0 is defined and times n is defined in terms of times n-1.
This is one possible definition of the class of functions that "boil down to" the successor function. If that's what you had in mind, then you can reword your question as, "Are all functions just primitive recursive functions?". Since this class of function has a name, you might have guessed that the answer is no.
Another class of functions called general recursive functions adds another way to specify a function, the "unbounded search operator" which returns the smallest natural number that satisfies a condition. For example, "the smallest prime bigger than x" would be a general recursive function.
It turns out that you can also define "the smallest prime bigger than x" in a primitive recursive way (it's just more complicated), so that function also happens to to be a primitive recursive function. However, there are some functions which are general recursive but are NOT primitive recursive. In particular, the Ackermann function. It is recursive, but there's no way to write it as a primitive recursion.
So now, another question we can ask is "Are all functions just general recursive functions?". Again, the answer is no. However! General recursive functions actually correspond with computable functions, functions for which an algorithm can be written. This correspondence lead in part to the Church-Turing thesis which I think you might find interesting.
But like I said there are some functions that are not general recursive, functions that are not computable. The most well-known is the busy beaver function.
Edit: One thing I want to point out. In programming, a "function" is an implementation of an algorithm. In that sense, every (programming) function is a general recursive (math) function. Every (programming) function can indeed be boiled down to constants, projections, and the successor function, by using composition, primitive recursion, and the unbounded search operator.
As for the busy beaver function, since there's no algorithm for it, it's not a function in the programming sense of the word. It's a function only in the mathematical sense that it's an abstract, but "well-defined", mapping from one set to another.