And when it can't, attempt to classify which subset of the integers can be reached. E.g. Legendre's three-square theorem.
3 is not enough by Legendre's three-square theorem.
The subsets reachable with 2 and 3 squares are fully characterized by Legendre's three-square theorem and
Compared to Waring's problem, this is potentially much harder, as we can go infinitely negative in our attempts, there isn't a bound on how many tries we can have for each number.
It is exactly what you'd expect from the name, Waring was watching Netflix with Goldbach, when they suddenly came up with this.