Not strictly a "bus" service through it's full history however First Aberdeen's Service 1 can be traced back as far as December 1883 (139 years ago) when it began as a horse drawn tram service operating from Market Street up Union Street and down Holburn Street to Bloomfield Road operated by Aberdeen District Tramways. By 1894 the service had been extended to operate from Bridge Of Don to Bridge Of Dee, the full route of which is still covered to this day. Aberdeen Corporation Tramways took over operation of the service in August 1898 and this is when the use of the Service number 1 first started along with the colour red on publicity, although the service was more commonly known simply as "The Bridges".
The service remained mostly unchanged until May 1958 when Aberdeen withdrew the last of its trams and the service was converted to bus operation. The bus service from 1958 followed the exact same route but was extended at either end to serve the new housing estate at Garthdee and the rapidly expanding housing developments north of the Bridge Of Don.
Below are links to two network maps:
Aberdeen Corporation Tramways (1898)
First Aberdeen (Current)