"Too many letters of recommendation" is not something I would consider a problem.
At some point additional letters may become overkill and not really help you that much. But I can't imagine them detracting from your application, provided that they are all good recommendations from high-profile people.
I would certainly not leave out a recommendation from a "big name" in your field because of volume. The fact that you have worked with a large number of high profile people, and they think highly of you, is a good thing. And you never know which of those people someone who is potentially hiring you might know or think highly of.
Five is certainly not too much if they want "at least 3".
The wording clearly means that more than three letters is acceptable. In fact, one might conclude that three letters is the bare minimum and a "good" application should have more.
I would normally consider five letters to be a lot (though not harmful, as discussed above). But it doesn't seem like an unusually high number if they want a minimum of three.