I think your training may have confused copyrights, patents, and trademarks.
Patent law allows triple damages in the case of willful infringement, and trademark allows triple damages in the case of a counterfeit mark that the infringer knew was counterfeit.
Copyright does have some things that can increase a damage award but there is no real triple damages mechanism.
The plaintiff in copyright gets a choice. They can ask for either:
• The actual damages plus the profits that the infringer made from the infringement. The latter is only to the extent that the infringer's profits exceeded the actual damages.
E.g., if you lost $100k due to infringement and the infringer made $70k your damages would be $100k. But if the infringer made $120k your damages would be $120k (the $100k you lost plus the $20k the infringer made over $100k).
• Statutory damages. It is often very hard to figure out actual damages so the law allows an alternative. If you elect statutory damages the damages range from $750 to $30000 per work infringed. The amount is determined by the judge or jury.
They can be decreased to as low as $200 if the infringer "was not aware and had no reason to believe that his or her acts constituted an infringement of copyright".
They can be increase up to $150000 if the infringement is found to be willful. This is the closest thing copyright as to triple damages.
The way registration affects all of this is:
• You have to register before filing a copyright lawsuit. There are some exceptions such as when a foreign copyright owner wants to sue over a work not published in the US but infringed in the US, due to Berne Convention requirements but we can ignore those here.
• You can only collect statutory damages for infringement that commences after registration (unless the registration is within 3 months of first publication).
• You can collect actual damages and infringer profits from infringement before registration.