Your understanding about lone pairs is wrong. All amines(primary, secondary and tertiary) have a single lone pair.
The difference in basicity is a mere effect of the inductive effect and steric effect.
The Inductive effect from the alkyl group increases its basicity(the larger/more branched the alkyl group is, the better). But bigger alkyl groups cause a lot of steric crowding of the lone pair, making it unavailable for attack. Striking a balance between the 2 results in the trend of basicity of the amine.
Examples:
tri-methylamine > di-methylamine > methylamine (Since Me group is rather small)
di-ethylamine > tri-ethylamine > ethyl amine (Since Et group is larger than Me, 3 Ethyl groups result in a higher steric effect than 3 Methyl groups)
Now considering a rather large alkyl group.
di-tert-butylamine > tert-butylamine > tri-tert-butylamine
here, tert-butyl group is rather large causing a lot of steric hinderance. hence in the case of tri-tert-butylamine, it causes the lone pair to not be available for attack, making it the least basic.