In the context of coded caching in the $K$-user BC, our work reveals the surprising fact that having multiple ($L$) transmitting antennas, dramatically ameliorates the long-standing subpacketization bottleneck of coded caching by reducing the required subpacketization to approximately its $L$th root, thus boosting the actual DoF by a multiplicative factor of up to $L$. In asymptotic terms, this reveals that as long as $L$ scales with the theoretical caching gain, then the full cumulative (multiplexing + full caching) gains are achieved with constant subpacketization. This is the first time, in any known setting, that unbounded caching gains appear under finite file-size constraints. The achieved caching gains here are up to $L$ times higher than any caching gains previously experienced in any single- or multi-antenna fully-connected setting, thus offering a multiplicative mitigation to a subpacketization problem that was previously known to hard-bound caching gains to small constants. The proposed scheme is practical and it works for all values of $K,L$ and all cache sizes. The scheme's gains show in practice: e.g. for $K=100$, when $L=1$ the theoretical caching gain of $G=10$, under the original coded caching algorithm, would have needed subpacketization $S_1 = \binom{K}{G}= \binom{100}{10} > 10^{13}$, while if extra transmitting antennas were added, the subpacketization was previously known to match or exceed $S_1$. Now for $L=5$, our scheme offers the theoretical (unconstrained) cumulative DoF $d_L = L+G = 5+10=15$, with subpacketization $S_L=\binom{K/L}{G/L} =\binom{100/5}{10/5} = 190$. The work extends to the multi-server and cache-aided IC settings, while the scheme's performance, given subpacketization $S_L=\binom{K/L}{G/L}$, is within a factor of 2 from the optimal linear sum-DoF.