  Fermi/LAT has detected long-lasting high-energy photons (>100 MeV) from
gamma-ray bursts (GRBs), with the highest energy photons reaching about 100
GeV. One proposed scenario is that they are produced by high-energy electrons
accelerated in GRB forward shocks via synchrotron radiation. We study the
maximum synchrotron photon energy in this scenario, considering the properties
of the microturbluence magnetic fields behind the shock, as revealed by recent
Particle-in-Cell simulations and theoretical analyses of relativistic
collisionless shocks. Due to the small-scale nature of the micro-turbulent
magnetic field, the Bohm acceleration approximation breaks down at such high
energies. This effect leads to a typical maximum synchrotron photon of a few
GeV at 100 s after the burst and this maximum synchrotron photon energy
decreases quickly with time. We show that the fast decrease of the maximum
synchrotron photon energy leads to a fast decay of the synchrotron flux. The
10-100 GeV photons detected after the prompt phase can not be produced by the
synchrotron mechanism. They could originate from the synchrotron self-Compton
emission of the early afterglow if the circum-burst density is sufficiently
large, or from the external inverse-Compton process in the presence of central
X-ray emission, such as X-ray flares and prompt high-latitude X-ray emission.
