How many pi bonds are there in bf3?

The quick answer is, “There are no in ##”BF”_3##”, but the answer isn’t quite that simple.
You can write four Lewis structures for ##”BF”_3##.
In one structure, the ##”B”## atom has 3 single bonds to the ##”F”## atoms and an incomplete octet.
The other three structures have double bonds to the ##”F”## atoms.
Most texts say that the double bond structures do not occur, because they put a positive charge on the ##”F”## atom and result in a separation of charge.
Therefore, ##”BF”_3## has no π bonds.
However, theory says that when you can write more than one Lewis structure for a molecule, the actual structure is a resonance hybrid of them all.
The double bonded structures may be minor contributors, but they still contribute a little bit.
In that sense, the ##”B-F”## bonds have some π character.
Furthermore, the length of the ##”B-F”## bonds is shorter than expected for single bonds.
This is consistent with some degree of π in ##”BF”_3##.