"Never" vs. "never ever"

Solution 1:

The difference is, as Kristina Lopez pointed out, a matter of "degree of emphasis." If you really wanted to emphasize your utter unwillingness to use a particular cup, you could add a comma after the word never, so as to slow the reader down and make him or her pause for effect.

That technique would also work well with the spoken word, if you were to raise slightly the volume of your voice on the word ever:

I never, ever use this cup.

There can then be no doubt in the reader's or listener's mind that you are loath to use that cup! (I'd feel the same way if the cup was used, for example, to scoop up poopy kitty litter! Yuck!)