How does viral DNA "force" the cell to make so many copies of itself in such little time? Is it some kind of super-promoter? Is there a type of virus that does not have this mechanism making it so that the cell only produces a few (perhaps only about 10, a very small number) viruses per day?

Thank you,

CDB

All of these are true. Viral promoters can be very strong. The CMV promoter from cytomegalovirus, for example, is commonly used in molecular biology because it is so strong. Other viruses and viral transcripts can have weaker promoters, or conditional promoters only turned on in certain circumstances. Latent viruses, when they are lying dormant in a host cell, would meet your condition of producing few-to-no viral particles per day.