Answer: No, it is not rational (it is irrational)
If we had something like 7.78787878... where the "78"s go on forever, then it would be rational. This is called a repeating decimal.
Something like 7.787887888... does not have a repeating pattern. It has 7's and 8's, but the number of 8's seems to increase by 1 each time you list a 7. So this is why the given number is not rational.