Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist? Does $1.0000000000\cdots 1$ (with  an infinite number of  $0$ in it) exist?

As @Lehs said, infinite means endless - every integer value $n$ is finite, so if you assume infinite sequence of 'zero' digits, then whatever number $n$ you think, the $n$-th position holds digit 0. Consequently there's no space where you can append 'one'.
