Why are amidines more basic than amines? Amidines can be thought of as carboxylic acid derivatives with the carbonyl oxygen replaced by a nitrogen and hydrogen atom; amidines exhibit conjugation.



Amidines are more basic (less acidic) than amines and amides by a factor of around ${10^{3}}$ and ${10^{12}}$, respectively (from pKas).

I would naturally think that conjugation would lower the energy level of MO containing the lone pair; thus, the lone pair would be more bound to the nucleus and less available to deprotonate. 

This is an argument similar to explaining why pyridine is less basic than piperidine; $\ce{N}$ is $sp^{3}$ hybridized in piperidine but $sp^{2}$ hybridized in pyridine. Since $sp^{2}$ orbitals have less p character than $sp^{3}$ orbitals (2/3 vs 3/4), the lone pair on $\ce{N}$ in pyridine is at a lower energy level than the lone pair on $\ce{N}$ in piperidien, and pyridine is less basic than pyridine. 



Although I realize that the positive charge on the amidine is stabilized by conjugation (as explored by a previous post comparing the basicity of amidines and amides), wouldn't my reasoning with MO theory outlined above suggest a lowering of the basicity of amidines?



References

Clayden, J., Greeves, N., Warren, S. Organic chemistry, 2nd ed.; Oxford University Press: New York, 2012.
Note that when you are comparing amidines with amines, you are comparing 2 nitrogen atoms with 1 unless you are comparing it with diamines.  Amidines which have 2 ionizable nitrogen atoms are stabilized  by 2 resonant structures, therefore making them more likely to accept protons (that is more basic).  Similarly, guanidine (not to be confused with guanine which is a nucleic base) which has 3 ionizable nitrogen atoms are stabilized by 3 resonant structures thereby making it even more basic. 
For additional reading, check out the basicity of the following drugs.
Betrixaban (which has 1 amidine group)
Pentamidine (which has 2 amidine groups)
Guanethidine (which has 1 guanidine group)
Metformin (a biguanide, that is, having 2 guanidine groups fused together)
Altretamine (which has 3 guanidine groups fused together into a cyclic structure)
Famotidine (which has 1 amidine group and 1 guanidine group).