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The record for the number of people in space at one time is thirteen, which has been set a few different times (as discussed in this question, this other question, and this article). In order:

  • March 1995, STS-67 (7), Mir (3), Soyuz TM-21 (3)
  • February 1997, STS-82 (7), Mir (3), Soyuz TM-25 (3)
  • July 2009, STS-127 (7), ISS Expedition 20 (6)
  • March 2010, STS-131 (7), ISS Expedition 23 (6)

At the moment, it peaks at nine (one ISS expedition plus one visiting Soyuz). When the US commercial-crew vehicles start flying, I assume they will be timed so that there is not a Soyuz and US crew rotation flight at the same time - it would seem to make things very complicated - and that will keep the peak at nine.

A Shenzhou flight at the same time as crew rotation would bring the total to twelve.

With what we know about future plans, is there any plausible combination of missions likely to fly in the next few years that would set a new record at fourteen?

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    $\begingroup$ Depending how you count space tourism, it may be very soon. If BO or Virgin Galactic start flying groups of 5-7 people, the amount of people in space concurrently could easily top 14. Maybe 2020? $\endgroup$
    – Dragongeek
    Nov 16 '19 at 22:07
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    $\begingroup$ SpaceX and Boeing will be doing direct handovers at least once, so the Soyuz crew of 3 + Boeing Dreamliner crew of 4 + SpaceX crew of 4 will get us 11 on the ISS. $\endgroup$
    – geoffc
    Nov 17 '19 at 0:32
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    $\begingroup$ @Dragongeek good catch! six on ISS plus seven on a flight would be thirteen, so we'd beat the record (for about half an hour) if there was an ISS crew rotation flight and a VG flight at the same time. I don't know what sort of frequency they're planning to fly with, though... $\endgroup$
    – Andrew
    Nov 18 '19 at 0:27
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This answer points out that there were 16 people in space (above 80 km) on July 11, 2021:

  • Seven people on the International Space Station: Three Americans, two Russians, and one from both France and Japan.
  • Three Chinese on the Tiangong Space Station.
  • Six people on Virgin Galactic's Unity, including Richard Branson, two pilots, and three Virgin Galactic employees.

There were exactly 14 people in space -- all above 100 km -- on July 20, 2021:

  • 7 on the ISS.
  • 3 on Tiangong.
  • 4 (including Jeff Bezos) on New Shepard 4.3.
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    $\begingroup$ Starman is still up there, somewhere, over the rainbow. $\endgroup$
    – Snow
    Jul 23 at 13:58
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    $\begingroup$ "in space (above 80 km)"... choose one. either "in space" or "above 80km". These are not equivalent statements.... Now the 20 july statistic is valid, as they crossed past the Kármán line at 100km. $\endgroup$
    – PcMan
    Jul 23 at 14:34
  • $\begingroup$ @PCMan The original answer refers to space and 80 km. DrSheldon is just repeating that information. Since there is debate about where the Karman line is located (which defines in space) it makes sense to clarify which definition of space is used for the July 20 feat. DrSheldon could have clarified by stating "7 on the ISS in space (above 350 km)", "4 on New Shepard (above 100 km)", and so on. :-) $\endgroup$
    – JohnHoltz
    Jul 23 at 17:45
  • $\begingroup$ I am happy to go with the McDowell approach and treat "space" as 80km, so both qualify - although I admit I am secretly waiting for there to be more than thirteen in orbit ;-) But good catch that the absolute record depends in part on definitions! $\endgroup$
    – Andrew
    Jul 23 at 19:32

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