There appears to be plenty of TF work available seeking small factors of 2^p  1 for p currently under consideration by GIMPS.
There is also plenty of work available to try to crack 2^p  1 which are known to be composite, but have no known prime factors; or remaining composite cofactors of partiallyfactored 2^p  1.
As a personal hobby, doing TF for larger p is unobjectionable, but that's about all you'll achieve. It wouldn't be of any use to GIMPS. For exponents of order 10^{12}, PRP or LL tests are out of the question for the foreseeable future.
Before looking at trilliondigit numbers for which all you'll ever be able to determine is whether they have really small factors, it might be nice to crack, say, 2^1277  1, a known composite of 385 decimal digits.
