Doubly charmed B decays with the LHCb experiment

$CP$ violation is a necessary ingredient to create a matter-antimatter asymmetry. However, the amount of $CP$ violation incorporated in the Standard Model is orders of magnitude too small to explain the observed asymmetry. This thesis documents two studies of $CP$ violation with doubly charmed $B$ decays. Both were performed using Run I data collected by the LHCb experiment in $pp$ collisions at the LHC. This corresponds to 1.0~fb$^{-1}$ collected at a centre-of-mass energy of 7~TeV and 2.0~fb$^{-1}$ collected at a centre-of-mass energy of 8 TeV. The first analysis is a search for $B_c^+$ decays to two charm mesons, $B_c^+\to D^{(*)+}_{(s)}D^{(*)}$, where the second charm meson is either $D^0$ or $\overline{D}{}^0$, which are sensitive to the CKM angle $\gamma$. No evidence for signal is found and limits are set on twelve decay modes. The second analysis is the measurement of the $CP$ asymmetry in $B^+\to D^+_{(s)} \overline{D}{}^0$ decays. The results are \begin{align*} \mathcal{A}^{CP}(B^+\to D^+_{s} \overline{D}{}^0) &= (-0.4\pm 0.5\pm 0.5)\%,\\ \mathcal{A}^{CP}(B^+\to D^+ \overline{D}{}^0) &= (+2.3\pm 2.7\pm 0.4)\%, \end{align*} where the first uncertainty is statistical and the second systematic. This is the first measurement of $\mathcal{A}^{CP}(B^+\to D^+_{s} \overline{D}{}^0)$ and the most precise measurement of $\mathcal{A}^{CP}(B^+\to D^+\overline{D}{}^0)$ to date. The results are consistent with $CP$ symmetry.