any R (methyl, ethyl, etc.) group apart from Benzene
Not exactly true. Vinyl (CH=CH
2), cinnamyl (CH=CH-Ph), and other such conjugated groups will also pull electron density away from nitrogen.
However is it not possibly also due to the fact that the electron density at the nitrogen in this molecule would be extremely high
It's high, but the electron density on the nitrogen of trimethylamine is not all that much greater than dimethylamine.
and so would repel the other electrons that would be coming from the methyl groups
No. The only effect that might have on an intramolecular front would be if there was some other incredibly electronegative group attached to one of the alkyl chains. The two electronegative groups would probably not associate with each other. To wit, alpha dicarbonyl compounds orient themselves such that the two oxygen atoms of the carbonyl groups point away from each other.
The pka trends among alkylated acids isn't that large of a gap. Alkyl groups just don't perturb the electron density map as much as, say, halogens or aryl groups:
compound pka
protonated ethylamine 10
protonated phenylamine (or protonated aniline) 4.6
protonated dimethylphenylamine (dimethylaniline) 5.2
protonated diphenylamine 0.78
Adding one aryl group drops the pka almost 5 units. Adding another aryl group drops it almost another 5. Adding 2 methyl groups only raises is 0.6 pka units.