Overheat would appear to be your main hurdle.
Two displays daisy-chained over Thunderbolt, leaving a third to run from HDMI has been achieved, but at a cost…
However, in more detailed evaluation, a helpful EveryMac.com reader reported:
I have a "Mid-2012" Retina Display MacBook Pro with two 27" Thunderbolt Displays daisy chained in clamshell mode. I tried adding a third 1920x1200 Dell monitor via the HDMI port, and it actually does work. But here is the catch: with no external displays, the system fans hover around 2600 RPM. With two 27" Thunderbolt Displays, regardless of what I do, the fans hover around 2800-3200 RPM. The GPU diode temp reads 69°C (156°F). If I force it via a fan control program to run at 2500 RPM, the GPU heats up. I was afraid of damaging my computer so I turned the fans back to auto-mode.
Now the kicker. I tried with the three monitors and the fans go up to 4800 RPM. That's without doing anything. If I start running a YouTube clip, or do anything remotely CPU or graphics intensive, the fans max out at 5900 RPM (which is the max the machine will do). I exceeded that by having a few videos running at the same time, fired up 3 virtual machines, and had it compile some code. The whole system became choppy and it began stuttering.
Since then, I've never used three monitors, just the two Apple Thunderbolt Displays. Apple probably doesn't officially recommend using three displays because it gets too hot.
The same helpful reader more recently followed up with:
I've recently replaced the two 27" Thunderbolt Displays with an LG 34UC97 Cineview Display, which has a 3440x1440 resolution. I tried this LG display with the "Mid-2012" 15-Inch Retina Display MacBook Pro and it works, but if I keep the MacBook Pro display open, the fans spin to 3500 RPM or so. When I power just the external display, it hovers around 2300 RPM.
I replaced the "Mid-2012" 15-Inch Retina Display MacBook Pro with a "Mid-2014" 13-Inch MacBook Pro -- specifically the MacBook Pro "Core i7" 2.8 13" -- and this model has no problems, thermal or otherwise, running both the external 3440x1440 display and the internal display at the same time, even though the external display is higher resolution than Apple officially supports for either of the two external displays.
It seems like the Iris 6100 graphics processor can handle a much higher resolution monitor, but the performance in 3D games and such is no match for the NVIDIA 650M in the "Mid-2012" Retina Display MacBook Pro.
Ultimately, although the Retina Display MacBook Pro models are technically capable of running three external displays as well as the internal one, the above reader is highly likely to be correct that the notebook can overheat with three connected.
Consequently, EveryMac.com does not recommend running more than two external displays on the Retina Display MacBook Pro models. However, it also is clear that some models can run a single external display that is a higher resolution than Apple officially supports for either of two external ones.