It is a well-known fact that the largest known prime number for several decades now has been a Mersenne prime, even though more and more of them have been found over the years and there have also been efforts to find other kinds of primes, like the one that proves that $10223$ is not a Sierpinski number.
But what about pseudoprimes? Does the largest known pseudoprime tend to be a Fermat pseudoprime to say, base $2$? Does the search for large prime numbers help reveal larger pseudoprimes?
There are some enormous Carmichael numbers (eg. 16+ million digits), pseudoprime to any coprime base, which can be generated by Löh & Niebuhr's method.
Update: this 2011 Hayman & Shallue poster reports the construction of a Carmichael number with more than 10 billion factors and around 295 billion decimal digits.