It's hard to answer that without knowing more of the situation, did you for example after the format install anything over top of the drive? What did you format the drive with? Have you been using the drive since then or did you immediately disconnect it? etc.
Typically for the right price you can restore/recover just about anything (at least partially) if you have a company who specializes in it perform the work and you've not overwritten it (and even sometimes then SOMETHING can be recovered.) That being said, the fees are enormous.
The secondary option is to purchase software that does automated recovery which is hit or miss and can be expensive.
The third option if you have enough technical acumen is to perform a recovery yourself by identifying file types (images, music, etc) by head/foot of the files and slicing them out using a data carver, but that requires a extremely high degree of skill and knowledge.
A lot of this relies on you not having overwritten data.