(This might have been asked before, but I could not find it.)
A lot of questions and answers on this site have some piece of code (to be compiled with one of the standard engines) and a screenshot. As many online services prove, it is very easy to get TeX running on a server. So why do we not have a compilation engine built into this site, that you can use to compile your document and save a screenshot to be used in the question? (note: only if you explicitly ask it to, see the edit below). Obviously, there are concerns, like which part of the document should be on the screenshot, but that is a matter of letting the author specify this. Also, there are questions and answers that will not be compatible with such a feature, and will still require manual compilation and screen-capture; but those are, IMO, a minority.
EDIT: How could it concretely work: There are several formats (plain, LaTeX, ConTeXt), several engines (pdfTeX, LuaTeX, XeTeX). We just need to support those; then it is up to the user to specify how they want their document to be compiled. A greater concern is that TeX changes, the packages get updated, and some of the old code on the site might not compile anymore. Well, this is obviously only an issue if we would like to ensure that people can recompile old code if they would like to update an old question. If we do want to ensure backwards compatiblity, then we might use TeX Live, and include the version from each year. Then the user could specify that they would like to use, say, the 2014 version. This will not work for all questions (like questions about extremely new versions of packages not yet included in the newest TeX Live version), but many of them. So in the majority of cases, I am confident the system would work; and for those that do not work, people can always stick to the good old manual solution.
EDIT: This question is not a duplicate of Automatic rendering of code provided in questions/answers?; that question (at least in its original statement) asks for a MathJax-like solution like the one found on Math.SE and Phys.SE. That was not what I referred to; rather, I want an engine to automatically compile MWEs using standard TeX engines. That being said, the question was modified to sound more like the present one. However, nobody answered the question in that form. In other words, that question did not "solve my problem".
EDIT: People appear to have gotten the wrong impression that the feature would be forced upon you, that it would compile your code even if you didn't want it to. This is not what I meant; it should only compile your code if you explicitly ask it to. That's what buttons are meant for.