The complexity of Karatsuba's algorithm, because it's a recursive one that gets "wider" at every level, is dominated by the width of that recursion. The top level always has a small number of operations, so we don't explicitly count them there, but the bottom has the truly huge number of operations that contribute to the algorithm's complexity, because each level of recursion dramatically increases the number of operations. Some number of those operations (most of them, in fact) will be single-digit additions.

Memoizing number-by-digit multiplication doesn't make multiplication O(1) because one must still do an N-digit addition (which is O(N)) for each digit.