The clear winner is the Tyrell family, prominent in Essex from soon after the Norman Conquest until the death of Sir John Tyrell in 1877. They were based at Heron Hall, East Horndon, and the church contains a number of their monuments; their house was demolished in 1788 (although some associated buildings remain), and latterly they lived at Boreham House. The first sheriff of the family was Sir John, sheriff of Essex and Herts in 1413–14 and again in 1422–3; he was also sheriff of Norfolk and Suffolk in 1426–7. He was one of the Duke of Gloucester’s retainers and fought at the Battle of Agincourt. His son Sir Thomas was sheriff in 1440–1, 1444–5, and 1459–60. Then in 1480–1 comes another Thomas Tyrell, and in 1502–3 and 1508–9 Sir Thomas’s son Humphrey. Edward Tyrell was sheriff in 1512–13, Sir Thomas in 1518, and Edward again (or another) in 1527–8.
Sir Henry Tyrell was sheriff in 1551–2, and after that they are more spread out: Sir Charles Tyrell Bt in 1695–6, Sir John Tyrell Bt in 1750, and John Tyrell of Boreham in 1770-1. The baronetcy had died out in 1766, but a new one was created in 1809 for John Tyrell of Boreham House, High Sheriff in 1827–8. The High Sheriff in 1887–8 was John Lionel Tufnell-Tyrell, also of Boreham House: his mother was a Tyrell, his father a Tufnell, thus joining the Tyrells to a family that, between 1785 and 1931, supplied four High Sheriffs.
That gives (I think) a grand total of sixteen for the combined families, twelve for the Tyrells alone, putting them head and shoulders above the other families in the Essex Top Ten.
[Many of the individuals mentioned here will be found in Essex Worthies: a biographical companion to the county by William Addison (1973).]