He did it: Redbull Stratos jump!

I'm not saying his acceleration is actually uniform, only that it sets an upper boundary to his maximum speed.
I hadn't considered the possibility that he'd fall at an angle though, that's an interesting approach.

Mach 1 is actually 343.2 m/s by the way.

Still for him to reach Mach 1.24 the angle would have to be:

2*36529/(259*cos(x)) = 1.24*343.2 => x = 48°

Yeah, he didn't fall at a 48° angle, nope.

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Oh I didn't actually bother to look up Mach 1, just took it from earlier posts in the thread...

At any rate, your approach still doesn't make sense. You can't look at the end points, find the average and say absolutely anything about what happens in between unless it's a linear equation.

If you want an absolute boundary use Vf^2=2ad and get 840m/s (assuming he falls straight down under sea level gravity). Might as well pretend he's a spherical cow in a vacuum.

You're not looking at what I wrote. The speed profile will always be above the linear one I'm using as the limit situation. For any other speed profile that accounts for the drag force the maximum speed will be closer to the average speed he reached which doesn't change, so his maximum speed is inferior or equal to the maximum speed calculated supposing it's a linear equation.

Let me write something clearer with images and graphs here.

So here we have two speed profiles, one is my limit case linear equation. The other is a case where drag force is considered, no real numbers of course.
I think you can agree with me that the downwards acceleration decreases as our friend Felix's speed increases, so the slope of the speed vs time curve should too. With that we'll always have a curve that starts out with a greater slope and ends with a smaller one, like the one in my example above.

With that it's easy to see that the curves considering the drag force will ALWAYS have an average that's closer to its maximum when compared to the linear profile.

Now, here I'm not actually considering what the real max speed he'd reach if there was no friction or calculating the effect of the drag force. What I'm doing is applying these theoretical profiles to the numbers we can trust in his story: the distance he fell and the time he took, which, together, get us his average speed during fall.
So if for both of the cases shown in the graphs Vavg has the same value, it should be obvious that Vmax for any case with the drag force taken into account will always be smaller than that of the linear one.

The speed needed to reach Mach 1 is lower at high altitudes though. 343.2 m/s is at sea level.

The speed needed to reach Mach 1 is lower at high altitudes though. 343.2 m/s is at sea level.

That has no bearing, since what they are saying is that he reached a maximum speed of 834mph (372.8 m/s) which is Mach 1.08 when considering Mach 1 = 343.2, that's still higher than what I've shown to be the maximum speed he could have had. The point at which he reached the higher numbers of Mach was most likely just before he opened his parachutes, at not so high of an altitude.

EDIT: So I've read some more on the story and found something that actually breaks my theories: he would have reached his top speed way before he opened his parachutes. Actually, there is a point of maximum speed while he's still at very high altitudes and after that the increasing viscosity of the atmosphere actually generates enough drag force to slow him down. Meaning that my supposition that his downwards acceleration was always positive is actually wrong.
If I still want to disprove the story I'd have to calculate the actual speed profile using the physical properties of the air at different altitudes (pressure and temperature taken into account) and estimate a drag coefficient. I'll have to take out my Unit Operations book.