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
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.