I played around with a multiplicative Fibonacci series once. pretty neat. I have no idea whether it approached Phi or not.
I guess I took artistic license to get it started:

1
1
2
2
4
8
32
256
8192
2,097,152
17,179,869,184
36,028,797,019,000,000
618,970,019,640,000,000,000,000,000


formerly known as etaoin...