I always wondered if some hidden pattern would be exposed when visualising numbers in unconventional ways in numbers with no known pattern such as Pi or prime numbers. A sort of multi-dimensional rendering that suddenly reveals a hidden pattern.
You can modify the continued fraction slightly to make pi regular as well, but the normal continued fraction sequence doesn't give much of an insight. Other than the fact that 3 + 1/(7 + 1/16)) is a damn good approximation (7 digits, pretty good for something that can be written using only 4 digits total: [3;7,16]).
But phi is indeed especially interesting because of what its sequence implies for rational approximations of phi.