The Windows Club forum member hackerman1 claims Windows 7 runs even on ancient systems like a Pentium II 266MHz with only 96MB RAM and a 4MB GPU. Unfortunately, he gives no information about the performance of the operating system on such a slow PC, he only says he got an insufficient memory error when trying to boot with only 64MB RAM memory. Next he's going to try a Pentium I 166MHz with a 1MB graphics card.
While the official Windows 7 requirements for installation are a 1GHz processor and 1GB of memory, hackerman1 has proven you really can get a way with less. A lot less in fact. His slowest successful test managed to get Windows 7 running on a Pentium II 266MHz proc with only 96MB of RAM and a 4MB GPU.
It seems that the limit came when he tried to go below 96MB of RAM, receiving an "insufficient memory" error when trying only 64MB of RAM.