There are two issues here.
First, you have to develop the programming skills that will allow you to succeed as a professional programmer.
Second, you want to get into the game industry.
Clearly, you do not need a degree to develop programming skills. However, a good college program may well be the most time efficient mechanism to get those skills; one of the drawbacks of being self-taught in anything is that by definition you are learning from someone who doesn't know what they are doing.
So, one way or another you need to develop professional grade skills. Could be a college program, could be the various online free college courses, could be sitting down and just working through it and reading a lot as you go.
Which brings us to the second issue. I'm assuming you actually want someone to pay you (since if you were doing it entirely on your own you wouldn't be worried about finding a job).
So, here is the problem. Games are one of the few parts of high tech (if not the only part) in which there is a glut of entry level talent. And the thing is, there isn't a simple "do this, then do that, then do this other thing, yay job!" list to give you. People get jobs without degrees. People get jobs without prior experience. People get jobs without demos. None of that helps you unless you are one of those people, and if you were, you wouldn't be asking this question.
It is entirely possible that last year a studio in your area hired someone who was almost exactly the same on paper as you are today. That doesn't mean you'd get a job there today, maybe another studio had a layoff and there are now people with professional experience on the market. It is entirely possible that the person tossing your resume in the trash bin had inferior credentials to yours when they got in to the industry. None of that matters.
If you really, seriously, truly want to get in, and you want to get in as a pro, you need to come across as the best qualified person for the job at the time they are making a hiring decision. For entry level, you want to have a degree, references, good networking, and a great demo or demos. Also, you want to be lucky.
So, do you need to go to College? No. But you do need the skills. And if all the programming you've ever done is some HTML, it is likely that formal instruction will get you the skills in a shorter amount of time than doing it on your own. Also, a good CS degree will help you if (as is likely) you end up leaving the game industry before you leave the workforce.