Integers proceed upward in an infinite sequence, so no number can be said to be the "highest," as it is always possible to add one and get a higher number. Infinity is generally not regarded as a number but as a concept. The highest named number is the googolplex.
A googol is 10^100, or a one with 100 zeros after it. This is 20 orders of magnitude higher than the estimated number of hydrogen atoms in the universe, which makes a googol difficult to use in real-world math. A googol to the googol power, a googolplex, is a one followed by 10^100 zeros. This number can be expressed as 10^10^100. If written in its entirety, however, the sheet of paper it is written on would not fit inside of the observable universe.