I personally like using fractions of 9.

  1/9 = 0.111...
  2/9 = 0.222...
  3/9 = 0.333...
  ...
  8/9 = 0.888...
  9/9 = 0.999...

What's neat is that this trick works for any repeating decimal, with any number of digits in the repeating part. For instance:

  123/999 = 0.123123123...
  999/999 = 0.999999999...

Multiply or divide by powers of 10 as necessary to shift the decimal point, and add the non-repeating part.

Once you accept this mapping, it's trivial to treat 0.999... as 9/9 (or 99/99, or 999/999, etc). Which can be simplified to 1.