I have an experience advantage of being able to ship working software that actually meets requirements. Unlike many younger devs.
The reason I ask is that I am a self-taught web developer (BA in English & Classical Studies) considering getting a master's degree in computer engineering. I am not certain if I do go down that road that I would want to stay in web development, but I am rather passionate about the web so I would not want to rule it out.
If you don't have a CS degree, getting a more technical degree couldn't hurt you, but it's more about the opportunity cost.
The masters degree is certainly not that useful for Web development, but it's a lot of fun and was pretty enjoyable to interact with smart people. It's also helped to have more formal learning in things like compilers for the few times I've needed to write a parser, etc.
There are absolutely jobs where it would help, like research. If you wanted to work on the super-technical stuff at Google or Facebook, like developing new languages, you'd probably need that MS.
But generally speaking, employers care exclusively about your experience (partially because there are so many self-taught people that do good work), and a Masters just says, "I'm going to demand a higher salary".
On a semi-related note, I'm self-taught, like you are, and I considered a Masters. I now laugh at that idea -- I don't know even a tiny fraction of what's needed to get a Masters. It was very humbling to talk to people with CS degrees and hear all the things they learned.
When you miss out on an undergraduate CS degree, you miss a lot of theory and math. Jumping right into an MS program would be incredibly difficult unless you're a prodigy of math and logic (and have the time to spend on catching up).
(None of that is to say that your typical web development job requires a CS degree at all. We use high-level languages and, oftentimes, widely-used, lightly-customized software.)
Not many people can (or do) write code until 60. Most will become managers. If you don't want to be run over by the next generation, develop and manage the resource that you have that they can't catch up with: experience.
You can develop that resource by doing a wide variety of work or by focusing on a niche. I've seen developers who only fix bugs in ____ language. Or you can work on a wide variety of projects, in a wide variety of roles, and do consulting.
Unfortunately, most coding is treated as a commodity by the business world, and the younger devs have a cost advantage over you.