I was wondering for what kind of commutative rings we can always construct an infinite descending chain of distinct prime ideals?
I doubt that there is a crisp definitive answer to your question, but here are a few implications and, maybe more importantly, non-implications.
a) If a ring is strongly infinite dimensional, it is infinite dimensional.
b) If a ring is noetherian it cannot be strongly infinite dimensional.
c) If a ring is infinite dimensional, it needn't be strongly infinite dimensional.
d) If a ring is non-noetherian it needn't be strongly infinite dimensional.
This is a correction of my previous post.
The statement "For a local ring $O$ to be infinite dimensional and to be strongly infinite dimensional are equivalent." seems to be wrong.
What remains of my previous post seems to be this: