No, 10^10^10^10^10 is not 1 googolplexplex. A googolplex is 10^10^100, which is 1 followed by a googol of zeroes. 10^10^10^10^10 is 1 followed by 10^10^10 zeroes, which is much smaller than a googolplex.
A googolplexplex is a hypothetical number that is 1 followed by a googolplex of zeroes. It is so large that it is difficult to even imagine. It would take a googolplex of years to write out the number in decimal notation.