What is a good word(s) for someone who excessively asks for information that they have no business knowing?
Solution 1:
In general, this person could be described as nosy. If you happen to be in Pittsburgh, PA, you could also call the person nebby. A good noun for this is busybody.
Solution 2:
Intrusive may qualify this person or his behavior.
Definition: annoying someone by interfering with their privacy; intruding where you are not wanted or welcome to thrust or bring in without invitation, permission, or welcome.
Solution 3:
Asking about such things excessively is being nosy.
Illicitly acquiring the answers to these questions (e.g. by rifling through other employee's private information without permission) is instead a violation of privacy.
Nosy is a derogatory term which, in a business context, may make it sound like you, rather than the employee, is at fault. From the viewpoint of a 'nosy' person, the other party is 'hiding something', just like how calling an employee a 'crybaby' may result in others wondering if in fact the issue is that you are 'insensitive'. Derogatory character trait terms like these tend to come in pairs. If you do feel the need to include a character-trait based derogatory term for this, "Intrusive" sounds better than "nosy" in formal contexts. You should really avoid both, though.
A violation of privacy is an event/action and taking grievance because of it makes your position sound neutral and supported. From the perspective of the privacy-violator, others will assume, your privacy didn't matter, and that claim is socially unacceptable. (It also provides actionable information the people you have a problem with can use to change their behavior in the future, but that's better left to workplace.se to discuss)
Solution 4:
In New Zealand (and Australia) there is a colloquial term "sticky-beak" which is used both as a verb and a noun. As in to stick ones beak where it does not belong.
i.e. "Luke is such a sticky-beak" or "Rachel, stopping being such a sticky-beak".
noun: sticky-beak
1. an inquisitive and prying person.
verb: sticky-beak
1. pry into other people's affairs. "I don't mean to stickybeak, but when is he going to leave?"