A bit of background on the anniversary being marked by the York County, ME half dollar of 1936...The 1936 York County, ME Tercentenary half dollar was issued "in commemoration of the three-hundredth anniversary of the founding of York County, Maine."
(US Public Law 74-822) One problem with this stated purpose lies in the fact that York County (originally Yorkshire County) was created on November 20, 1652 not in 1636.
The area that today constitutes Maine was granted to Sir Ferdinando Gorges and John Mason by King James in 1622; the grant made no provisions for a formal government in the area. In 1629, the land was divided, with Mason assuming control of the western territory that would become New Hampshire and Gorges getting would would become Maine.
In 1636, the year marked by the coin, there was not yet a York County (or Yorkshire County)! The present-day county's ties to 1636 rest on the establishment of a local court of commissioners that first convened in Saco in that year and the fact that the town (now city) of Saco is located within the present-day borders of York County, ME. This "government" did not become officially recognized until 1639, however, when King Charles I granted such governing authority to Sir Fernando Gorges, the then sole title holder/owner of the land (via Charles' previous royal grant). The area defined in the grant was officially referred to as the Province of Maine.
Side Note: Ferdinando Gorges never visited his Maine grant. He sent family members to handle matters for him while he stayed in England.In its early days, the Province of Maine was a part of the Massachusetts Colony; MA took control of Maine in 1647 and it remained a part of MA until Maine became a state in 1820. Massachusetts was formally granted authority over the area by King Charles; the Massachusetts Colonial Government established Yorkshire Country in the Province in 1652. Yorkshire County was larger than present-day York County, as several counties were carved/formed from the original over the years. The name of the County honors James, Duke of York, who went on to become James II, King of England, Scotland and Ireland in 1685.
Early/rudimentary settlements are often promoted as the original roots of a town/city/county, well before legal documents (e.g., incorporation) are drawn up. So, while calling 1936 the 300th anniversary of the founding of York County is a bit of a stretch, there was at least an actual local event in 1636 (i.e., formation of the Court of Commissioners) with which the coin's sponsors could associate themselves. This initial governing body gives rise to the county's claim of continuous government since 1636 (even if it's not truly continuous York County government).
1936 York County, ME Tercentenary

For more on the history of the York County half dollar, see:
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1936 York County, ME Tercentenary-
1936 York County, ME Tercentenary - Revisited-
Quick Bits #14 - 1936 York County Half Dollar-
What If? 1937 York County, MaineFor other of my posts about commemorative coins and medals, see:
Commems Collection.