There is no such number.
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There is an infinite number of possibilities to the highest number.
There is no highest number
There is no highest number, highest prime number, or highest composite number. Therefore, there can't be a second-highest, third-highest, etc., of any of these types of numbers, either.
numbers never end, so it's impossible to say what the highest number is. I'm not sure that we even have names for all the place values yet. :D (10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000) haha, is there a place value for that i wonder???
There is no highest square number. If there was such a number then that number squared would be a higher square number!