Converting any computable function into a sequence of additions
I don't quite see the connection between the preamble and the question.

Regarding the three fundamental operations, this is of course an arbitrary selection. On one hand, you could say that the most fundamental operation is the unary successor function $x \mapsto x+1$. On the other hand, you have hyperoperations, some of them occurring naturally in Ramsey theory and combinatorial pseudorandomness.

Regarding your actual question, the answer is plainly no. No finite sequence of addition operations can compute the function $(x,y) \mapsto xy$, for example. Indeed, the output of any given sequence of additions on inputs $x,y$ is $O(x+y)$, which grows too slowly.

This is a special instance of a more general theorem about nested bounded iteration, which states that a function $f(n)$ can be computed using $m$ nested bounded loops iff it is recursive and eventually bounded by $A(n,m)$ (the binary Ackermann function). This is part of the circle of ideas around primitive recursive functions.