This post will prove an upper bound of $1/(4-2 \sqrt{2}) \approx 0.854$, improving slightly on Christian Remlings $0.87$.
Consider $M < N < 2M$. At time $M$, there are $M$ intervals, each of length $\geq \mu/M$. Let $S$ be the number of those intervals which are subdivided at time $N$, we have $S \leq N-M$. So $M-S$ intervals remain undivided. We have $$(M-S) \frac{\mu}{M} + (N-M+S) \frac{\mu}{N} \leq 1.$$ Letting $S$ go all the way up to $N-M$ makes $\mu$ as large as possible; in this case we have $$(2M-N)\frac{\mu}{M} + (2N-2M) \frac{\mu}{M} \leq 1.$$ Taking $N = \sqrt{2} M$, we get $$(4-2 \sqrt{2}) \mu \leq 1 \ \mbox{so} \ \mu \leq 1/(4-2 \sqrt{2}).$$