In Modern Day British, Austrailian and US language the biggest number with a name is "Unvigintillion" or 1 followed by 66 zeros.
However in very old British and Austrailian language it is still "Unvigintillion" but it refers to 1 followed by 126 zeros.
Source Wiktionary.
The 3rd biggest number with a name is a googol: a "1" followed by 100 zeros.
The 2nd biggest number with a name is a googolplex: a "1" followed by a googol zeroes.
The BIGGEST number in the world is a googolplexian: a "1" followed by a googolplex zeros.
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There is no largest number. For a finite number, call it "n", it is always possible to add 1, to get a larger number, "n + 1". For an infinite number, adding one would NOT give a larger number, but if you calculate 2n you get a number that is larger than the original number, even if the original number is infinite.