The largest named number is a googolplexian, and the largest known prime number is known as the 39th Mersenne prime. There is no distinct largest number. Numbers continue on forever, as explained by the idea of infinity. Infinity itself is not considered a number and can't be measured.
A googol is the number 1 followed by 100 zeros. The much larger googolplex is the number 1 followed by a googol of zeros. The googoplexian is the largest named number, and it is the number 1 followed by a googoplex of zeros. The 39th Mersenne prime is 2 to the 13,466,917 exponential minus 1. This number is 4,053,946 digits long.