It seems 5 doesn't work, as it turns out 4 of them are actually similar. But 7 works. The idea is to try to form a spiral of rectangles from corner to center.
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Let's start with a sqrt(7) square. The first rectangle starts from any corner. It has sides a=(8sqrt(7)-sqrt(133))/21 = 0.458736 and b=21/(8sqrt(7)-sqrt(133))=2.1799. The next rectangle sits at the adjacent corner. It has sides c=sqrt(7)-b and d=1/c. We can continue in this fashion.
The first rectangle could have been a=8sqrt(7)+sqrt(133))/21 = 1.55707 and b=21/(8sqrt(7)+sqrt(133))=0.64223. The spiral can be formed similarly.