Covering one square by three smaller squares

If the small squares are allowed to overlap it is possible.

I don't have any graphics software handy so this is going to be tricky.

If the big square is ABCD, place a small square EFGH such that E=A and FG passes through B. Let M be the intersection of GH and BC. Using 3-4-5 triangles you can show that BM=0.3125.

Similarly, place a small square IJKL such that I=A and JK passes through D. If N is the intersection of KL and CD then again DN=0.3125.

Since CM and CN are both less than 1, the remaining space can be covered by the third square.

Here is a really awful hand drawn image, but perhaps it's more helpful than the description. . .

I can't compete with David's picture, but you might find Squares Covering Squares (part of Erich's Packing Center) useful. In particular it shows that Henry Dudeney found (in 1931) a covering not unlike David's which allows three unit squares to cover a square of side length $\sqrt{\frac{1+\sqrt5}{2}}\approx1.27202$.

In particular this answers Ivo Beckers' question: the minimum side length to cover a square of side 1.25 is $\sqrt{\frac{5\sqrt5+5}{8}}\approx0.982689222,$ assuming the optimality of Dudeney's construction.