The "Birthday" of OpenStreetMap is lost in the sands of time. Observance of the anniversary of the creation of OpenStreetMap is held on or about the 9th of August, which is the anniversary of the registration of the OpenStreetMap.org domain name. The concept of OpenStreetMap predates the domain name registration, but that seems a suitable anniversary date.
In August this year, we're 16!
Details of this year's events: