Numbers Expressible as Sum of Five Distinct Squares

Theorem
The largest positive integer which cannot be expressed as the sum of no more than $5$ distinct squares is $188$.

Both $188$ and $124$ require as many as $6$ distinct squares to represent them:

Proof
From Numbers not Sum of Distinct Squares, the following positive integers cannot be expressed as the sum of distinct squares at all:
 * $2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128$