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Power was doubling every second. Not a linear increase.

Sure, and we know why. Running at 200 MW got them into what is known variously as Iodine Valley or the Iodine Well or Xenon Poisoning. Within the fuel rods this stuff normally burns off in the neutron flux in normal operation. At low power levels, though, it builds up. Both Iodine 135 and Xenon 135 will decay by themselves in time (half lives of 8 and 12 hours respectively IIRC, although I didn't look that up). But while they're still there both have large neutron capture cross-sections and soak up the neutrons that would normally be moving between the fuel rods. And bringing a reactor in that situation up again by pulling the control rods means that nothing happens for a while, which tempted them to bring them up still further, and so they did. The exponential increase in power was essentially a reflection of the iodine and xenon burning off within the fuel rods, and when it finally did so the reactor was running wide open. It gets hot faster than humans can react and faster than the control rods can descend. They did get the rods moving, but as you say, it made things worse.

This was compounded by the lack of real-time measurement. The actual heat distribution within the reactor was only visible after the equivalent of a 286 computer crunched the input numbers and printed the profile out on a dot-matrix printer. That process meant that the operators were always 10 minutes behind what was actually happening. Minutes for a decision, seconds to move the rods, heat increase in milliseconds. "Unstable" is an understatement.

My sources for this, if anyone cares, are James Mahaffey's Atomic Accidents and Grigori Medvedev's Truth About Chernobyl, (formerly "Chernobyl Notebook"). Great reads, both of them. Medvedev was the guy who was flown in to clean up the mess.

That all make loads of sense. They had shut down the reactor and were quickly powering it back up because the senior engineer wanted to get the damn test done. They had pulled the rods to do a fast power-up. . . not a smart move but that’s what he ordered, again against the line-engineer’s best training. He was in a rush to get the test done at his intuition that it could be done at 200 MW instead of the protocol’s 700 MW-1000 MW.

The two younger line engineers, fairly new from training, were too timid to stand up to the older guy with his years of experience and Party standing. They were arguing for a timed re-start and then test with protocols. Not going to happen according to the senior engineer. That would probably take them into the next shift. . . and he wouldn’t be in charge; the plant chief engineer would be. Ironically, that guy had no nuclear training at all. He was appointed due to Party standing and his experience with electrical plants.

One thing I found interesting was that they had no way of measuring core temperature that deep in the reactor. Their sensors were all higher up, and they did not have sensors down where the reaction was out of control during this incident and they were not aware of it. The upper levels were seemingly normal until they weren’t. It sounds as if someone made the assumption that all areas would be the same and management could again cut corners to save rubles on sensor costs.