I’m planning on applying to Stanford REA this fall, and I’d like to know how important having a great additional rec is on an application to this calibre of school?
That is, do those who apply without additional rec have about the same admission rate as those who apply with an additional rec?
I know people probably don’t have access to this kind of info but any insight is appreciated 
There are no stats on that. Certainly a great extra LOR would be a positive addition to any application – but it is one of many many components. But unless you can get an additional non-teacher letter of recommendation (ex. boss, clergy, etc.) who can truly shed some light on a quality(s) that will not be covered in a teacher recommendation then the general rule is to not submit an extra letter – admissions officers are not looking for multiple recommendations which say essentially the same thing.
Second what @happy1 said. If you’re a competitive applicant at a school like Stanford, your application will stand on its own without any extra letters.
If there’s not a strong personal connection with the person that’s writing the additional rec, you risk looking like you’re trying to use the person’s stats to get in instead of your own. However, if you feel that the third rec will show an additional aspect of who you are, then go ahead!
Totally agree. Otherwise, the thicker the file, the thicker the kid.