What should be done about this employee who we've already countered once?
People generally don't leave jobs only because of money (though in your specific case, there might be an exception). Location, benefits, work environment, work content, people, etc, all affect this too.
When you offer someone more money, you only remove the "reasons I want to leave" if the only driving factor is money. Generally this isn't the case. And sometimes, even if it is, it can breed discontentment if the employee thinks "wow, they were underpaying me by X and only offered it when I quit!"
This person is incredibly unlikely to stick around in the future. Trying to quit twice in 5 months is not a good sign they will stick around.
Oh, and it's entirely possible the employee realized they are valuable and wants to be paid more now rather than later. Or is making this all up, realizing you guys offered ridiculous "stay please!" packages and might want more.
Regardless of which of the above it is, it's not in your best interests to keep the employee.
How to make sense of "holy wow you underpay!"
People are taking offense to the "yeah but he's still underpaid!"
There are a ton of factors which can affect this. Let's consider the following example. The employees work for a small company and are software engineers and the OP is making $60k/year. According to glassdoor, this is right around the low end of software engineer salaries in the United States (it goes from 61k - 115k). For some startups this might not be unreasonable and for many it's probably high. Often in startups you won't get a raise without asking, either.
The employee now also has some experience after a few years and is likely making the same $60k starting. Perhaps the company is making more money now and the employee has proven their value. After this the company offers the employee $90k to stick around.
Now, the employee is realizing their value and skills and starts applying to a few larger companies for senior level positions and gets offered $115k/year. Maybe they applied to Google, this seems pretty reasonable as their average salary is about that.
The employee now has a job offer which is nearly double their current $60k salary. Are they completely underpaid at their startup in this example?
Of course, it's also possible the company is paying $40k, too, and they are getting screwed. But jumping to this conclusion is not necessarily the only possibility.