Thank you for trusting your question to JA today. I am a licensed attorney with over a decade of law practice and over 20 years of experience in the legal field. I’m happy to be of assistance.
Unfortunate, it will be very unlikely that you can ever enter the military again. While an RE-3 can receive a waiver, waivers are not a right and they aren't being granted right now because there is no shortage of recruits trying to join.
Additionally, depression is a very serious concern for the military, as suicide is still a major factor in the deaths of American service members.
To be able to get back in, you have to get that record changed in regards ***** ***** RE-3 and the only way to do that is to file a DD Form 149 with the military records corrections board, in an effort to get them to change the record regarding your separation. You would have to show that the record was either in error or was an injustice. Now, you certainly have some arguments to that effect. The issues that your Drill Sergeant was facing are certainly telling of an unprofessional individual and so it is not unreasonable to suspect that you were being harassed. You do, however, have this medical history concerning depression. You will need medical evidence to support that this was an event based depression and not a physiology based depression...evidence sufficient to convince the board to change your RE code and narrative for your separation to make it easier to reenlist.
This is the only conceivable method of reentry.
If you have any further questions, please let me know. I invite follow up questions, so use REPLY for those. If you have no further questions then good luck going forward and please do not forget to rate my service with a top-three rating so that I receive credit for working with you today. Also, feel free to request me in the future, if you have questions concerning a different matter.