If you're looking for the asymptotic behavior, it may be of interest to approximate this process as a product of uniform random distribution. 

The product of $t$ uniform random variable has probability density

$$p_t(x) = \frac{~|\log x|^{t-1}}{(t-1)!}$$

In particular

$$\int_{0}^{1/n} p_t(x)~\textrm{d}x = \frac{\Gamma(t,\log n)}{\Gamma(t)}$$ 

*might* be a decent estimate of $P(x_t=1)$

For instance, $P(x_2=1) \simeq \frac{\Gamma(2,\log n)}{\Gamma(2)} = \frac{\log n}{n} + \frac{1}{n} + O(\frac{1}{n^2})$... not quite $\frac{\log n}{n} + \frac{\gamma}{n} + O(\frac{1}{n^2})$ but not bad either.

In general

$$P(x_t=1) \simeq\frac{1}{n}\sum_{k=0}^{t-1} \frac{\log^k n }{k!}$$