Ordinarily, we would predict the order to be C > N > O > F.

C is the least electronegative atom.
It has the least tendency to hold on to its electrons, so it is the best electron donor and the strongest base.
So, why is C out of order in your series?
We are comparing "CH"_3^-CH−3 and "CH"_3"C≡C"^-CH3C≡C−.
The difference is in the hybridization of the C atom.
In "CH"_3^-CH−3, the hybridization is sp^3sp3.
In "CH"_3"C≡C"^-CH3C≡C−, the hybridization is spsp.

An sp^3sp3 orbital has 25 % ss character, while an spsp orbital has 50 % ss character.
ss electrons are held more tightly to the nucleus, so an alkyne C atom is less basic than an alkane carbon.
The order of acidities (with "p"K_"a"pKa values) is
"CH"_4CH4 (48) < "NH"_3NH3 (38) < "CH"_3"C≡CH"CH3C≡CH (25) < "CH"_3"CH"_2"OH"CH3CH2OH (16) < "HF"HF (3)
So the order of basicities is
"CH"_3^-CH−3 > "NH"_2^-NH−2 > "CH"_3"C≡C"^-CH3C≡C− > "CH"_3"CH"_2"O"^-CH3CH2O− > "F"^-F−