There is no biggest number. Numbers go on forever. Infinity isn't a number.
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Numbers DO go on forever, but the question is what is the biggest number that human beings have named so far?
No. (Googolplex plus one) is bigger. As you can see, there's no such thing as a "biggest" number. No matter how big a number you name, I can always name a bigger number simply by adding '1' to yours.
1111
No. ( 1021 + 1 ) is bigger. There's no such thing as the biggest number. Whatever number you name, no matter how big it is, I can always add ' 1 ' to it and make a bigger number.
The biggest number that can be possibly be written, is 10^4*185
Since there is no biggest integer, there can be no biggest rational number - since any integer is a rational number with the denominator = 1.