I like trennag's bluesy quilt! Another pattern that uses just squares is flying squares.
I don't think you have to do a lot of cutting to make a neat quilt. I just noticed that one of my mom's quilts has a giant star block (Kansas star, maybe?) as the center medallion -- the square pieces are 5"-6" so the whole thing is easily 2 feet square.
Here is my "just lay 'em out and sew 'em together" quilt:
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