Do you have a portfolio ?
For what kind of position would you apply?
When in university, i attended a 3D Graphics Programming course (more of a primer). For the evaluation we would be able to choose between various projects, one the teacher encouraged us to take was to join the community of an open-source 3D engine and develop something on their to-do list.
How does it sounds?
If you are more of gameplay programmer you can join a mod or indie project (that is employing the same tech as the company you would like to join).
I mean, I'm no programmer, but if you are trying to be creative and come up with an original concept you are doomed, you have to waste a lot of time minding things that aren't really a concern for a programmer when hired.
Joining a project will free you of this kind of problems and will also display your ability to work in an established team (which is what you'll be end up doing anyway).
It's a win-win... win to me, if your added value will, say, help get to a release in a few months from now.
Quote from RxanaduDoes anyone think this may be useful?
What do you mean? In terms of being able to land a professional job later on? Of course. If you have experience in solo projects, mods or internships then you'll be one step ahead of your competition.
Companies hire interns not because of charity but because they want free junior positions to say it rather bluntly. They won't expect the same kind of finesse/experience as a junior position (but there's a lot of competition even for internships so it comes rather close) and they'll give you training so you'll get better. They hire interns cos there's a job to be done, so ideally they would want to hire you as a junior after the internship to fill up that need, IF you end up delivering on your potential (if they hired you as an intern they must've seen some kind of potential). Which explains why they'd put in the effort to train you in the first place. So that becomes a win for both the studio and the intern (studio filled the work, intern got a job).
It's (sadly) not unheard of that smaller studios keep on hiring new interns endlessly so they don't spend as much resources (money), but shooting themselves in the foot in the long term... Cos they don't end up with an experienced staff.
So in a TL;DR way: Interns are expected to be as close to the real deal as it gets. Companies don't like to waste time and money on a person. You basically have to think what you can give back to the studio you're applying for.
On a side note, are you in a college/university? Cos it's gonna be a tad harder to get an internship if it's not set up by a course, at least for some companies. I think it has to do with some kind of benefits companies get from the government for hiring students, etc.
Also: Never say you never finish stuff, you want people to think you're incompetent? Sorry to come of so harsh but I mean, those are your words. It's better to make small prototypes instead of whole games if you can't get it done then. Show code for those small things and get them optimized. You're never going to make "a full game" anyways, as a coder you'll be put on tasks so prove you can focus on some features. In the best case, part of a bigger unit, like a mod. Team experience is valued very highly, game dev (like most jobs) is kind of a social job. A lot of people depend on you and you'll have to communicate with different departments and people. But again, how much depends from studio to studio.
I hope that helps! 
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