I accidentally forgot to extend my star player and had to arb him a third time, and now he won't sign a long term contract "because I went to arbitration three times." He has 99 patience. If I trade him do you think the receiving team would be able to re-sign him? Trying to figure out how much value to expect in return (if the receiving team is able to re-sign him he'd have A LOT of value).

No way to know that I am aware of. Id say the chance is slim but have no real proof.

I traded for Pablo Profar several seasons back, after his previous owner missed the arbitration period in his final season. He was unwilling to sign an extension with them, but he did sign a new deal with me. Given that he received a max contract as a 33 year-old, it seems I saved $30 million in salary over the life of his contract. It's rare, but this is/was the best player in Shtickless history (14 AS games, 12 SS, 7 MVPs by age 34), so I'd never say never regarding extensions, and you can try to get full value on your guy.

I'd never say never... I'd also never say always. I think historically, overall, many owners here would have an expectation that the player would be willing to re-sign with his new team.
However, my expectation would have been that a 99 Patience player would overlook a third arb. So I think the takeaway is that there's no absolute guarantee.

I traded for a superstar pitcher who wouldn't sign for his prior owner, and then, he wouldn't sign for me either. Lost him in free agency.

What is the cut off for signing a second long term contract?

Well, the guy re-signed. Quite a bargain for the receiving team, but I also got more than I would’ve had I held him and received a type A.

https://www.whatifsports.com/hbd/Pages/Popups/PlayerRatings.aspx?pid=8825478