A search for decays of $B_c^+$ mesons to two charm mesons is performed for
the first time using data corresponding to an integrated luminosity of 3.0
fb$^{-1}$, collected by the LHCb experiment in $pp$ collisions at
centre-of-mass energies of 7 and 8 TeV. The decays considered are $B_c^+\to
D^{(*)+}_{(s)} \overline{D}^{(*)0}$ and $B_c^+\to D^{(*)+}_{(s)} D^{(*)0}$,
which are normalised to high-yield $B^+\to D^+_{(s)} \overline{D}^0$ decays. No
evidence for a signal is found and limits are set on twelve $B_c^+$ decay
modes.
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