  We make a detailed study on the photoproduction mechanism of the doubly heavy
flavored $B_c^{(*)}$ meson at the purposed Large Hadron Electron Collider
(LHeC) within the framework of nonrelativistic QCD. In addition to the
photoproduction mechanism via the gluon-induced channel $\gamma + g \to
B_c^{(*)} + b + \bar{c}$, the extrinsic heavy quark mechanism via the two
channels $\gamma + c \to B_c^{(*)} + b $ and $\gamma + \bar{b} \to B_c^{(*)} +
\bar{c}$ has also been studied. Those two extrinsic channels are generally
suppressed by the heavy quark distribution functions in proton, which however
provide significant contributions in low and intermediate $p_T$ region. A
detailed comparison of those channels together with the theoretical
uncertainties has been presented. By summing up all the mentioned
photoproduction channels, we observe that about $(1.04^{+0.90}_{-0.53})\times
10^{5}$ $B_c$ and $(4.86^{+3.72}_{-2.30})\times 10^{5}$ $B_c^*$ events can be
generated at the LHeC in one operation year with the proton-electron collision
energy $\sqrt{S}=1.30$ TeV and the luminosity ${\mathcal L}\simeq 10^{33}$
${\rm cm}^{-2} {\rm s}^{-1}$. Here the errors are for $m_c=1.50\pm0.20$ GeV and
$m_b=4.9\pm0.40$ GeV. Thus, in addition to the hadronic experiments, the LHeC
shall provide another helpful platform for studying the $B_c^{(*)}$ meson
properties, especially to test the extrinsic heavy quark mechanism.
