The first one is correct and would be the most obvious correct answer. Any more (two words) could mean no longer or no more in the second phrase. This could be confusing to the listener and could come across as you not agreeing with them.

I couldn't agree more.

I couldn't agree more. -you're definitely right. -that's definitely true. -in agreement. I couldn't agree any more. -maybe you used to agree with someone's opinion or request but you can't do it anymore for some reason. -not any more.

I know that the first one is definitely correct. If the second one is wrong I don't know.