There is good reason, based on circumstantial evidence, to believe that Enoch Fuller (1784 - 1860) was a direct descendant of Edward Fuller of the Mayflower Pilgrims, via his son Samuel. I have not, however, managed to find a single shred of documentary evidence with which to prove who the father of Enoch Fuller was.
Here are the family legends and the facts that we have today:
1. Family legend: Enoch's daughter Julina Fuller (1822 - 1880), an early student at Oberlin College and a pious woman by all accounts, told her ten children (including my great-grandfather) that, through her, they were Mayflower Pilgrim descendants.
2. Family legend: Josiah Fuller (1920 - 2005), son of Nathaniel (1896 - 1985), son of George P. (1844 - 1896), son of Josiah (1810 - 1895), son of Enoch, told his son the following:
"This is the story of our family as I have been able to put it together. Heather Makechnie, a genealogist I met at Proctor (Academy, Andover, N.H.) at my fiftieth reunion came up with this scenario by matching birth dates and places. Benjamin Fuller came with the first settlers to Colchester, New London County, Connecticut. He was one of the first persons to die in the settlement in 1740. He and his wife, Content, had three sons, Matthew, Josiah born 1729, and Benjamin. Josiah married Irena Dickinson in Sharon, Connecticut in 1749. They either moved to Dutchess County, New York or there was a boundary change. This location was called the Oblong, a disputed boundary area between the Hudson and the Connecticut rivers claimed by New York and the New England Colonies. Josiah and Irena had a son Josiah born in 1763. Josiah is listed as the head of a household in Columbia or Albany County in the 1790 census. Josiah had a son Enoch born in 1784 who joined the Western Reserve Company as a hired hand as they walked the Mohawk trail to Ohio. He came to Ashtabula in 1807. He died June 9, 1860, age 76 years in Saybrook, Ohio."
3. Fact: The name "Josiah" (Part 1) -- Enoch named his firstborn son "Josiah", and that Josiah also named his firstborn son "Josiah".
4. Fact: Enoch's daughter Julia Ann (1814 - 1894) married "her cousin, Aaron Fuller, of New York" (documented in Civil War pension application of Julia Ann Fuller). Aaron Fuller's father was "Josiah Fuller of New York" (documented in Ashtabula County death record for Aaron Fuller).
5. Fact: The name "Josiah" (Part 2) -- in the letters written back and forth between Raymond and Randall Taylor (grandsons of Julina), one of them, in discussing Julina's ancestry, referred to "Josiah Fuller who beat the snare drum in the Revolution."
6. Fact: Documented Mayflower Pilgrim descendant Josiah Fuller (b. 1729) who married Irena Dickinson and lived at Amenia (now Dutchess County, NY) had a son Varsel Fuller who served in the Revolution. Two of the affidavits supporting Varsel's pension application were by his brother, Josiah Fuller. This Josiah Fuller is not well documented, except by his affidavits for his brother Varsel. But we know he lived to advanced age, since the affidavits were in 1833, and one of them gives his age as 67. This Josiah appears to have lived at Saratoga, New York, continuously beginning with the 1800 census, and he appears to be the same Josiah Fuller found in the 1790 census at Hillsdale in Columbia County, NY. Several images from the Varsel Fuller pension file are shown below, along with my transcription of a Josiah Fuller deposition.