EDIT: As it turns out, I'm not the first or even the second person to run calculations like this:
Meeus' work (second link) mentions the 13425 CE event in "Table 1. Simultaneous and near-simultaneous transits of Mercury and Venus, years 1 to 300,000"
Within the limits of DE431 (7 May 13201 BCE to 7 May 17091 CE), there is no
time at which both Mercury and Venus transit the Sun.
The closest we get to this:
On 16 Sep 13425 CE at 11:57pm UTC, Venus starts transiting the Sun. This
transit ends the next morning (17 Sep 13425 CE) at 7:30am.
Less than 9 hours later, at 4:27pm, Mercury starts transiting the
Sun. This transit ends at 10:26pm.
The program I used to compute this:
The list of transits I computed while solving this:
Although I believe this answer is correct, Stellarium does not agree with
me, and HORIZONS doesn't compute positions past 9999 CE, so don't put too
much faith in this answer, since there's no good way to confirm it. I
believe that I'm correct and Stellarium is wrong this far in the future, but
it could be the other way around.
Even if my calculations are correct, the uncertainty in calculating the
relevant positions (Sun, Merucry, Venus, Earth) this far in the future is
high. On their own transit pages, NASA only computes Venus transits from
2000 BCE to 4000 CE, and Mercury transits from 1601 CE to 2300 CE, even
though they could've made the same calculations I made from 13201 BCE to
This suggests NASA isn't confident enough of Mercury/Venus (and Earth/Sun)
positions to predict that far in the past or future, so my results may be fairly inaccurate.