1. Literally, to give one a check with the amount left blank, so that it can be written out for the desired total. Mom gave a blank check to me so I could fill it out when I bought my school supplies.

2. By extension, to give one the freedom or permission to spend as much money or use as much resources as needed in pursuit of a desire or goal. The company was having a difficult time retaining good employees so the CEO gave a blank check to the manager to increase salaries and restructure his department. After the fifth murder, the mayor gave a blank check to the city's police officers to use whatever resources they needed in order to find the serial killer before he struck again.