The trends of bases of the type NR3, where R is an alkyl group or hydrogen, can be explained quite well, and quite simply.
The first point to note is that the strength of a base of this type is related to the availability of the electron pair on the nitrogen atom (as it is this pair that will attack any protons). As we know, alkyl groups are inductively electron donating, which we might expect to destabilize the nitrogen electron pair, and therefore make it more ready to attack protons, hence a stronger base. Therefore, we would expect that the trend of base strengths would follow the pattern:
However, this is not found to be the case. The basicity does indeed increase from ammonia through the monoalkyl substituted molecule to the dialkyl substituted molecule, but we find that the trialkyl version is actually less basic than the dialkyl.
For example, if we replace the R with methyl group to make it a real example, we find that the trend is:
What we have to explain here is why the introduction of one methyl group increases the basicity considerably, and the next has little effect, and the third actually decreases the basicity.
The addition of the alkyl groups does indeed increase the electron density on the nitrogen through inductive donation, however, there is a competing factor at work too: solvation energy. The ordering according to solvation energy is shown below:
Clearly, this order runs in reverse to the inductive donation, and therefore what we se overall is a balance of the two effects, resulting in the dialkyl base being the strongest.