18 centuries of Roman Empire
1790 years, to be exact. The Roman Empire began in 338 BC, when the city of Rome (a republic) imposed its direct rule upon the former league of Latin cities. It ended in 1453, as the last Roman Emperor, Constantine XI, died defending the walls of New Rome against the Turks. Not to my knowledge has any other state had a continuous existence for this long; the nearest would perhaps be Japan (arguably since about 400 A.D.).
Over these 18 centuries the fortunes of the Empire have waxed and waned greatly. However, for about the middle 3/4 of this time, from 146 BC to 1204 AD, it was without doubt the greatest and wealthiest power in the northern Mediterranean. Below are some small maps which give an idea of the Empire's changing territorial extent (including client states) over this period of greatness.