In line with how you were able to understand the Monty Hall problem, one key to understanding this issue is realizing that:
infinity + 1 = infinity
I think the hurdle here is: "0.0001 is between 0.999 and 1, so for every additional 9 you put after that decimal point, I can put another 01 thus making a number between 0.999... and 1." This turns into a race to a non-existent finish line.
Personally, picturing the digits never did it for me. I think I work better when thinking about quantities and values: the infinite number of 9s means the value will continue increasing until it gets to 1. Or just understanding that 0.000...001, an "infinitesimal" quantity, is the same as 0. Something like that.
infinity + 1 = infinity
I think the hurdle here is: "0.0001 is between 0.999 and 1, so for every additional 9 you put after that decimal point, I can put another 01 thus making a number between 0.999... and 1." This turns into a race to a non-existent finish line.