Does the black body radiation send the energy back out?
Exactly, this is the issue. If an object is able to absorb sunlight, it is also able to emit blackbody radition back towards the sun. When the temperature limit is reached, these two exactly cancel each other. The object will emit blackbody radiation with the same brightness as the surface of the sun.
Another way to look at it is to imagine yourself standing at the center of the concentrated sunlight and looking out towards the concentrator. The concentrator makes the sun look "bigger" from your perspective, and this is what makes the sunlight concentrated. The limit to this effect is if the sun fills all directions in the whole hemisphere above you. Now it will be as if you are standing on the surface of the sun, and all you can see in any direction is sunlight. Normally, the solar disc fills 1/45000th of the hemisphere above you here on earth, thus the limit of 45000 suns concentration.
But I could build up a lot of solar panels and use the electricity to heat up an oven more than the surface of the sun, right? Is that "cheating" in terms of thermo dynamics?
Yes, this would be like using a hydroelectric dam to power a fountain that sprays higher than the initial reservoir. Machines can convert a large amount of low-quality energy into a small amount of high-quality energy, even when passive components (e.g. mirrors or pipes) cannot.
What about a material that absorbs photons and emitts them at a higher energy level (emitting one after absorbing two)?
The most efficient possible way to convert sunlight to electricity is ~86% and is related to the second law of thermodynamics. So we use the heat flow from a hot reservoir (sun) to a cold reservoir (earth) and are able to convert some of that heat into work (electricity) which can then be used to heat something else to a higher temperature without breaking the second law.
ps. It's like making a VPN tunnel to the surface of the sun :-)
Wow, I love that! Great analogy!