When I was working with kW class lasers I did a lot of research to find the best vendors for highly reflective mirrors.... that had high damage thresholds...

The best I could find claimed 99.98-99.99% reflectivity, (I went with VLOC and MLD) MLD has worked with LIGO (the gravity wave detector) to improve their reflectivity a lot....

There are no "perfect" mirrors to my knowledge.

You can certainly make a cavity that has high reflectivity to increase the average power of a laser... but there are limits...

As for a sphere that has "perfect reflectivity" well by definition nothing would happen- no melting no explosion because that can only happen if absorption takes place, if there is absorption then by definition it isnt a "perfect" reflector....

In reality as intensities increase (assuming we are talking about real materials here) non linear processes would take hold an the whole thing would become a ball of plasma....

Assume for arguments sake that a person in an evacuated sphere that was 100% reflective turned on a flash light what would happen? Well, if the person ran the light until the battery ran out then the light would grow in intensity until the total power of photons equaled what the battery/flashlight could put out.... Hardly lethal....