There isn’t one- the numbers just keep going on forever. You can prove this by noting that for every finite number you can always add one to find a bigger number. Interestingly though – not all infinities are the same. For example, while the number of integers (like 1,2,3, …) equals the number of rational numbers (those that can be expressed as one integer divided by another integer), both of these infinite sets is smaller than the set or rational numbers (of which pi is an example). Infinity is a strange concept!
Theoretically there is no ‘highest’ number, as there can always be one higher – hence why we use the term ‘infinity’ to explain that it just goes on and on forever.
There are some interesting names for large numbers though. A ‘googol’ which is 1 followed by 100 zeros. You then have a ‘googolplex’ which is 1 followed by a googol zeros!