If you set your wallet to pay zero transaction fee what happens? Is there more than one possibility.
It makes sense that no transaction fee means that you get no priority, but will the transaction always be processed eventually? If yes, what is the maximum time frame? If no, is there a maximum time limit before it bounces back? Could it be stuck in limbo for months?
It will be confirmed eventually. If we consider the priority of the bitcoin day (1 bitcoin, 144 (1d) confirmations, 1 input, 1 output) and that its linear (.1 BTC, 1440 confirmations) the highest number of confirmations you would have to sit out in order to get a reasonable high priority would be for a 1000 Satoshi TX. A TX spending less than 546 satoshi is considered dust and would not get relayed by most nodes. This input would need 100,000 days or 14,400,000 confirmations to get a reasonable high priority for a confirmation without fee.
Depending on your wallet software or service the TX will get constantly rebroadcasted to make sure no nodes forget about it.
If you have a low fee and you cant stop the rebroadcasting the TX might be "in limbo" for months, yes.
With this said, I've sent transactions of 0.01+ before and they seem to confirm in a decent amount of time. I just did it for testing and it was back in 2013 or so, though. I haven't done any lately like that. I also sent a transaction for like 3 mBTC at one point and forgot to add a fee. It took ~37 hours to confirm.