US copyright act: "the copyright owner may elect, at any time before final judgment is rendered, to recover, instead of actual damages and profits, an award of statutory damages for all infringements involved in the action".
Statutory damages in the US are a minimum of $200 per work and a max of $150k. The minimum is quite the minimum too: "In a case where the infringer sustains the burden of proving, and the court finds, that such infringer was not aware and had no reason to believe that his or her acts constituted an infringement of copyright, the court in its discretion may reduce the award of statutory damages to a sum of not less than $200".
s. 504 "Remedies for infringement: Damages and profits" http://www.copyright.gov/title17/92chap5.pdf
Also can get lawyer fees under s. 505. So statutory minimum + big legal costs.