> Is there a limit to the amount of heat a body of space can transmit via black-body radiation?

There's a limit on the rate, rather than the amount, given by the Stefan-Boltzmann law. There will be grey body corrections for the heat shield.

Some details: https://en.wikipedia.org/wiki/Black_body#Radiative_cooling

Some even more gory details (slide 21 gives a sort of grey body curve; slides 108-113 are directly relevant; slides 34, 36 & 37 form a handy quick reference; slide 86 has a couple of graphs about foams):

http://www.ltas-vis.ulg.ac.be/cmsms/uploads/File/2015_10_21_...

and already that's more than I will ever really want to know, but this is probably of interest as a stepping stone for other HN readers. :-)

> how much heat an area of space near a star can have added to it

The region of space near a sun-like star generally has heat passing through it, starwards->infinity. Not much sticks around, and certainly not for very long.

There are limits on how much heat can be in the outer atmosphere of stars can be; the important thing is that "heat" here involves the presence of an amount substance, as well as how the average bit of substance in the region is moving in relation to other bits of substance in the same region ("temperature"). The limits are complicated because matter will tend to be blown away by the flux of radiation from the star through the outer atmosphere. The Eddington Limit is relevant here; Eddington's equation describes how radiation drives winds through a stellar atmosphere, to the limit where it blows the atmosphere out "to infinity".

The outer atmospheres of stars with atypically strong magnetic fields can be Super-Eddington, as can those around compact objects like neutron stars. Black hole accretion discs can also be Super-Eddington and the matter in the inner portions may get hot enough to disintegrate into gamma rays. This is called a "[big] blue bump", and implies temperatures of a hundred megakelvins to a few hundred gigakelvins or so (quasars have the very hot bits around them), although the temperature in much of the disc will struggle to reach a megakelvin.

> Is it infinite? Is the limit so high that it's far beyond even the atmosphere of a sun?

We've seen the "yes" answer to the second question.

The first is straight forward: high heat ~ high energy, and if you put enough energy into a small enough volume, it collapses into a black hole. It's easier to do this with a large amount of relatively low-temperature matter instead of a smaller amount of much higher-temperature matter.

> And are there special areas of space where the heat limit has reached its maximum and if so what does that mean for the properties of that space?

Black holes probably exist, given observations to date. Stellar mass ones are very cold. We have no evidence for black holes much smaller than our sun. They would be warmer, and would become very hot for extremely low-mass black holes. Here cold and warm relate to the very blackbody-like spectrum of the Hawking Radiation they emit. (~ nanokelvins or less. As said above, the accretion disc material can have much higher temperature).

The deepest layers of neutron stars are extreeeeemely hot (~ terakelvins). So are the outer layers. If they collapse, that heat is locked up within the black hole.

Pair-instability supernovae are the next hottest thing you can have, probably. In those, it's so hot in the core that the light produced as nuclei bump into each other is heavy in gamma radiation; a little hotter and you get gamma rays hot enough to turn right back into electron-positron pairs. (~ tens of gigakelvins). The heavy outer layers of the star then crash inwards as they lose their support from the outward pressure of the light (back to Eddington again). Kaboom! The Kaboom is likely so massive that not enough matter is left in the vicinity of the former star to leave a remnant like a neutron star or black hole. The matter thrown out of such a supernova can have ridiculously high temperatures, but rapidly become sparse enough that the heat per cubic metre drops off to nearly nothing. Hot protons arrive at Earth from such explosions as very-high energy cosmic rays, and when detectors pick them up lots of astronomers will get paged.

Penultimately some things which have very high temperature, but not much heat (because there's not much matter at that temperature; it's stray wispy sparse particles with lots of empty space between them. The products of smashed-together lead ions at the LHC is in exakelvins. Depending on model, the temperature of dark matter in active galactic nuclei can be in zetakelvins.
The daughter products of the highest-energy cosmic rays smashing into atoms in our atmosphere can be in yottakelvins. If we could collapse a spherical shell of photons into a black hole (a "kugelblitz"), the final temperature before the black hole appeared would be on the scale of the Planck temperature, meaning hundreds of millions of yottakelvins.

Finally, the extremely early universe probably had regions of higher temperatures still. Nothing we know prevents the temperature of the big bang from being infinite. However, nearly everyone hopes that quantum gravity would abolish that infinity by e.g. gravitational radiation undergoes a phase change in dense ultra-high but finite temperature regions, kind-of like how pair-instability supernovae's innermost pressures drop at their temperature maximum when super-hot gamma rays change into electron-positron pairs.