Google isn't a number.
The search engine Google used that name as it is a pun on Googol which is a number, and is 10¹⁰⁰ which is a 1 followed by 100 zeros
10¹⁰⁰ + 1 (a 1 followed by 99 zeros followed by another 1), 10¹⁰⁰ + 2 (a 1 followed by 99 zeros followed by a 2), 10¹⁰⁰ + 3 (a 1 followed by 99 zeros followed by a 3), ... are bigger than a googol, so yes, there are numbers bigger than a googol.
A googolplex is 10^googol = 10^(10¹⁰⁰) = 10^(100...0) where there are 100 zeros in the power of the 10; it is a 1 followed by a googol (ie 10¹⁰⁰) zeros.
In fact there are infinitely many numbers greater than a googol as the counting numbers (1, 2, 3, ...) can each be added to a googol, and there are infinitely many counting numbers.
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I don't think that there is a number bigger than its square as you are timesing the number Not true. Any number between 0 and 1 is bigger than its square.
Not one with a name!
Any number bigger than 7.37 and smaller than 8.43 will do that. There are an infinite number of them.
There is only one way to change a decimal number to a number bigger than it. Any number such as 5821.2347 adding a zero to either end won't make it bigger. The only way is to get a number negative or smaller than 1 but not 0 than making it to the power of 0 can make it one which is bigger.
A half can be any size, depending on what number you are halving. If your asking is 0.52 bigger than a half of one then yes it is.