For 40 years, it's been an unsolved question whether squares of size 1 to 38 can fit in a size 138 square. It's the lowest unresolved value in sequence A005842 Here's a packing where squares 10 to 38 are placed.
Is it possible to place squares 9 to 38 without overlap? Could 8 be added to that? Can all the squares be placed? This looks like a resolvable problem.
Sequence A0081287, packing sequential squares in the smallest possible rectangle, is unsolved above 32 squares.