There is no largest number because the set of real numbers is infinite. One of the largest numbers with a name is googolplex, which is 10 to the power of 10 to the power of 100.
Not only are the real numbers infinite, but they are uncountably infinite, which means there are more members of the set of real numbers than of any set of natural numbers. The natural numbers start with 0 or 1, depending on the definition, and continue with the rest of the positive counting numbers (1, 2, 3, ...). The real numbers consist of the rational numbers that can be written as fractions with integers for the numerator and denominator and irrational numbers that cannot be written as fractions in this way.