[nly]

Another intuition:

All the decimals that recur are fractions with a denominator of 9.

E.g. 0.1111.... is 1/9

0.7777.... is 7/9

It therefore stands to reason that 0.99999.... is 9/9, which is 1

[hoosieree]

That's a good one! Might replace my current favorite which is:

Let x = 0.99...

Then 10*x = 9.99...

And if we subtract x from both sides, we get:

10x - x = 9.99... - x

And since we already defined x=0.99... when we subtract it from 9.99..., we get

9x = 9

So we can finally divide both sides by 9:

x = 1