When ACGME does a full blown site visit, or even their annual review, of a program, they look at the case logs. That is one way they figure out if a program is providing an adequate amount of cases to train you. So, for instance, if everyone in your class only logged the minimum number, then ACGME may decide that there are too many people in the program for the amount of cases, and state that there has to be a decrease. (Have heard of it happening) It is also a requirement that the PD review your case logs quarterly, to make sure that they are done and appropriate.
Also, I have been requested to supply copies of case logs (I keep a copy of the grad's) in order for a grad to be credentialed (privileged) for the most mundane things, (placement of central lines), as well as maybe more specific (blocks, etc). I give the grads a printed copy of their logs, but I know that sometimes those get lost.
And don't CT Fellows have to supply logs when they go for their special boards? I know in years past they have.
In other words, get into the habit now. It will serve you well in the future.