We show that relativistic bremsstrahlung and inverse Compton scattering of radio-emitting electrons can easily account for the observed gamma-ray spectrum of 2EG J1857+0118 if the field strength in the shell is below similar to 30 mu G. This source is located at the eastern border of the composite SNR W44, where the expanding radio shell is interacting with a dense molecular cloud. The nondetection of this remnant above 250 GeV implies a cutoff or steepening in the electron spectrum above similar to 100 GeV. The E-1.66 spectrum of this radio/gamma-ray-emitting electron component is too hat to have its origin in standard first-order Fermi acceleration, but electron injection into the shell by the pulsar PSR B1853+01 over the 2 x 10(4) yr lifetime may explain why the Crab-like radio spectrum (S-nu proportional to nu(-0.33)) is about the hardest of all shell-type remnants. The injected energy would be sufficient to account for the required energy of 6 x 10(49) ergs if the initial spin-down power of PSR B1853+01 was about 10 times larger than the present spin-down power of the Crab pulsar. A steeper Fermi electron component may be present, but the observational data are not constraining enough to provide a meaningful limit on the presence of an additional similar to E-2 shell-type electron component. The predicted gamma-ray contribution from high-energy proton-gas interactions is about 20% of the observed EGRET flux above 100 MeV, which confirms our conclusion that the gamma-ray emission from W44 is dominated by a leptonic component.