Now, let’s get things straight, there is no such thing as a highest possible number. No matter what number you have, there is always a larger one. For example, you can always add 1 to your number and get a larger number)
The question really should be “what is the largest number that anyone has ever given a name to?”
So, the largest number that has a commonly-known name is a “googleplex”, which is a 1 followed by a googol zeros, where a “googol” is a 1 followed by 100 zeros.
However, there would be nothing to stop you from coming up with an even larger number, such as a googleplex plus 1, which is why this is such a complicated question.
In summary, then: the mathematical question “what is the highest possible number” has no answer, because there is no such thing. But the sociological question “what is the largest number that anyone has ever decided to give a specific name to, a name which has become commonly known” is, for now, a “googleplex” (until someone decides to coin a phrase for a still larger number and it catches on and becomes commonly known).