> I'm doing something wrong.
Back when I flunked EE, it was required to show your calculations. So don't say "did not come up with" and expect any help. How can we possibly know where you went wrong?
This one is worth knowing from Basic Inspection, because 97% of all inputs are wired this way.
First: the second stage is almost always wired Unity Gain. So ignore it.
Next: while (their) R2 and R5 kinda "must" be equal, it is useful to write both.
Recall that a happy op-amp has zero voltage between its inputs.
What makes this one happy?
Apply an input to the two external pins. The voltage on R1 200R "must" be the same. How can R2 have voltage? Current flowing to it from R2 R5. Why is there current? Because the output nodes of the two opamps will (we hope) assume a proper voltage.
Opamp inputs take zero current. (Not dead-zero, but usually utterly negligible for gain computation.)
Therefore R2, R2, and R5 all have the same current.
Assume we put 1V in. Assume the amp gets happy and 1V appears across R2 200R. This is 1V/200= 5mA. This must flow in R1=10K and R5=10K. 5mA in 10K is 50V. So the differential output must be 50V+1V+50V, or 101V. The circuit gain is 101V/1V= 101.
Another point to note. R2 200R has thermal noise similar to a 200 ohm dynamic mike. Even if the chip could be dead-quiet, noise figure is 3dB. If you want even higher gains, you do not {EDIT}increase R2 R5, you decrease R1. And if you want huge dynamic range, you decrease R2 R5 to 5K or maybe 2K (limited by how much strain the chip will stand). It seems reasonable, for low-noise design, to take R1 down to more like 5 ohms. However pots have large and unpredictiable wiper and hop-on resistances. Some of the glitching you hear near max gain is pot imperfection. Also, as shown, the DC gain is equal to audio gain. Using old rule of thumb, a 5mV DC input error is a whopping 505mV error at the output. A cap in series with R1 fixes this, but it should be bipolar, and for 20Hz at 5 ohms it must be like 2,000uFd. And now between pot and giant cap, the stray inductance in the "R1 arm" may bite you at the top of the audio band.