Is there any info on how far into the Sahara the Romans conquered, exactly?

Bit of a funny question, and depends on what you count as conquered. The Garamantians were subdued at one point by Septimius Severus, which is a good chunk inland.

I know there was the one exploratory expedition where they reached the other side. (Or was that Carthage?)

There’s actually an excellent wiki article on just that