What's the largest practical stable gain possible?
What is the largest single-stage/single-pole wide-bandwidth gain/GBW
  attainable (I seem to recall that a gain of ~200 is about the limit of
  discrete electronics).


These chopping amplifiers are the highest gains I've seen, at 240dB or higher with a unity gain of 300kHz. Thats a GBWP of 3*10^29, Don't get too excited, they are expensive. 


  What is the largest dual-stage gain/GBW attainable (most op amps would
  fall into this category). Could a multi-stage (>3) amplifier be compensated in such a 
  way to make this possible? 


IMO the main problem will be noise, especially if the op amps have a high gain feedback network. The problem is the noise of the first will be multiplied by the second, so if you have a gain of 10^6 on the second stage and a 1uVpp noise on the first stage, the second stage will see 1Vpp of noise from the first. 1uVpp isn't going to be useful for most designs.  

In all of the high gain systems I've designed, it's best to take care of most of the gain in the first stage. Even resistors become major noise sources at 1MΩ, and if you want a gain of "only" 10^6. 1/F noise is an even larger problem with high gain.

But really, what are you trying to do? Typical analog digital systems have a pre amp that feeds into an ADC. If the noise is gained up, one will only be increasing the number of noisy bits on an ADC. Even if the preamp is not being used for digital conversion, noise still applies in control systems. In my experience it's really SNR that matters, so if that figure hasn't been calculated then it should be.

There are multi stage options, on the preamp stage, the amplifiers can be paralleled for lower noise, with the noise being \$\sqrt(2)N_{amps}\$. Typically with normal op-amps gains higher than 10000 are difficult. I have tried paralleling op amps for lower noise, it works well. 


  Would a different topology (e.g., Norton or Transconductance
  amplifier) provide a way around these limitations?


Not that I know of, the way to get around high gain problems is most likely chopping amplifiers which help lower noise. The learning curve is very steep however. 

Another way to get around these problems is to divide up the bandwidth into different signal chains. For example: you need 1e13 with a bandwidth of 1e6, which means a gain of 1e7 at DC. One way to divide this up would be to have a multi stage signal chain with bandpass filters. One chain could cover 100Hz with a gain of 1e7 then another stage could cover 100Hz to 1000Hz with a gain of 1e7, the next would cover 1kHz to 1000kHz ect. This is only an illustration. 

Whatever you do, it is going to be both costly in both hardware and time to design. Now would be a good time to look at the system level requirements and make sure that they are correct and the cost of this design is justified. 

The largest practical stable closed loop gains possible (V/V) that are useful would IMO be around 10000.