The valence electron configurations of several atoms are shown. how many bonds can each atom make without hybridization?

 Boron has only three valence electrons and at best could form three bonds, not four. But in its ground state, there is only one unpaired electron, so it would make only one bond. 

 Boron always forms three bonds. Hybridization is not something that an atom "decides" to do or not do. Hybridization is a way to explain what we observe. And we observe three bonds for B in a trigonal planar arrangement, and this is consistent with sp2 hybridization. 

Likewise, you could say, that since N has three unpaired electrons that it will form three bonds "before" hybridizing. Again hybridization is a way to explain the bonding in N. Nitrogen is either sp, sp2 or sp3 depending on the number of pi bonds (where there are unhybridized p-orbitals). 

Following this same , oxygen form two bonds since it has two unpaired electrons. , that is the case for oxygen. But the 4 electron pairs that will be around an oxygen in a compound will either be sp2 or sp3 hybridized .