We present tradeoffs between RMR complexity and memory word size for recoverable mutual exclusion (RME) algorithms using arbitrary synchronization primitives. Assuming that each memory location stores $w$ bits, we show that $n$-process mutual exclusion has an RMR complexity of at least $\Omega\left(\min\left\{\log_w n,\log n/\log\log n\right\}\right)$ on the DSM and the CC model. For $w=(\log n)^{\Omega(1)}$, our lower bound asymptotically matches an upper bound by Katzan and Morrison (2020), whose RME mutual exclusion algorithm employs $w$-bit fetch-and-add operations. Our lower bound is the first one that does not restrict the type of atomic operations that can be executed on a memory location.