Canadian moving to Bucharest

Well, one way to stay longer would be to get a job. It'll be pretty hard, because companies have to prove that they specifically need your skills and they haven't been able to find a qualified person in the EU. Once you've found an employer, you can get a residence permit and a work visa.

Alternatively, find someone who is willing to employ you on paper, even if they don't actually need you. That'd cost them some money as even if they didn't actually pay you a salary, they'd have to pay taxes/contributions on your behalf, and you'd obviously have to refund them. It's an expensive way to go about it and also totally illegal, so I don't recommend it.

Another way might be to start up a company yourself, appoint yourself the adminstrator, and obtain a residence permit that way. Starting up a company isn't too expensive but I believe (check it out with a lawyer) that you now have to prove you have money to invest, something to the tune of 100'000 Euro. So you'd have to put that much money in your company account every time you wanted to extend the permit. You'd also incur running costs (taxes, accountancy fees, etc) even if the company effectively did nothing.

The last option (and some might say the most expensive!) would be to marry your Romanian girlfriend As the spouse of an EU citizen you would essentially have the same rights as an EU citizen, so you'd be able to work here more freely too.

Marry the girl.

Another way is to go out of the country every three months and re-enter ... Bulgaria is very near and not too far to drive to .... just a suggestion.  This again is not totally legal but it does work.