Would a "Triple AES" (in the sense of how Triple Des works) serve for a dramatic increase in safety?
It really depends on what sort of break AES would suffer. The primary issue with DES was that it's key length was too small (56-bits). Multiple encryption can help here because it increases the effective key length of the whole operation. The meet-in-the-middle attack on DES takes about 2^112 operations, which is infeasible to brute force anytime soon.

AES doesn't have an issue with keysize, so multiple encryption won't really help you that much in that sense. It really depends on what sort of attacks emerge on AES so it's hard to tell if multiple encryption will be better. It might be more secure, but it won't be a "huge" improvement and it's hard to judge.