In your sentence you need a double that to make the meaning clear.
Note first that the word that is often elided without impacting on the meaning, so that:
You'll discover that something doesn't exist
becomes:
You'll discover something doesn't exist
However, if you elide the second that in your sentence, you do change the meaning. At present your sentence is saying:
...you'll discover that the particular something concerned doesn't exist.
If you leave out the second that, the meaning changes to:
...you'll discover that something doesn't exist.
This second sentence fails to clarify which something you are referring to.
http://partofspeech.org/what-part-of-speech-is-that/