The meet in the middle attack reduces the time from 2^(2n) to 2^(n+1). This is why 3DES (even with 168 bits of keying) only effectively gives 112 bits of security. That's why I didn't propose 4DES, which wouldn't have any improvement in security over 3DES. However, my proposed 6DES would give 168 bits of security, or 8DES sould give 224, etc. 6DES has the advantage that you can use an existing 3DES implementation twice.
However, that's just the time complexity. The meet in the middle attack also requires storage for 2^n blocks, which is obviously not available for n=112, let alone larger values of n.