  We consider the repulsive Hubbard model on three highly frustrated
one-dimensional lattices -- sawtooth chain and two kagom\'{e} chains -- with
completely dispersionless (flat) lowest single-electron bands. We construct the
complete manifold of {\em exact many-electron} ground states at low electron
fillings and calculate the degeneracy of these states. As a result, we obtain
closed-form expressions for low-temperature thermodynamic quantities around a
particular value of the chemical potential $\mu_0$. We discuss specific
features of thermodynamic quantities of these ground-state ensembles such as
residual entropy, an extra low-temperature peak in the specific heat, and the
existence of ferromagnetism and paramagnetism. We confirm our analytical
results by comparison with exact diagonalization data for finite systems.
