When seeking out the finest academic editing services, one should avoid sliding into the pitfall of simply correcting typos. Academic editing serves to achieve clarity, flow, and logic while considering stringent rules like the APA, MLA, or Chicago styles.
I'd recommend starting with editors who take on academic work. Seek those who are able to handle dissertations, theses, or research papers, as they are familiar with complex academic writing. Sites like LinkedIn or freelance words work to connect you with independent editors. Just don't fail to ask for a sample edit, as that will gauge all sorts of things, like their attention to detail and style of communication.
If you work with limited deadlines, with a right turn-around not going over 24-48 hours and at a price so cheap, you'd think it was a sham. Well, a capable editor will not only fix grammar and formatting, but will also call your attention to certain areas where arguments could be much stronger or clearer.
Another piece of advice is to clarify your expectations. Do you need developmental editing (restructuring) from editing for grammar and consistency? The more specific you are about what you need, the better they will deliver.
Finally, if money is tight, there are graduate student editors or university writing centers. Most of them are fairly reasonable but give good service. Just remember that a polished professional writing piece will make the difference, so make that investment in someone you can trust!