This is a story I read sometime in the late 1980s or so, about a man who had perfect recall of every aspect of every moment of his life. I must have read it in either a science fiction magazine or a collection of short stories. (If I had to choose, I would say a magazine.) The publication date was probably sometime 1965 to 1980, but it may have been earlier.
The story is told in the first person, and one detail that stands out is his discussion about how he never knows whether other people to whom he's been introduced will remember him or not. There's a specific reference to how he can place every face he's ever seen, with an example of passing someone that had sat near him in the stands of a baseball game years ago.
The story may have used the word "eidetic" to describe his memory. (Note: This is not true, per the provided answer; perhaps I just associated this term with the story when I first learned it.) I think it was something he was born with -- not the result of an experiment or anything like that. I also think he did his best to hide his talent, to avoid the notice of government researchers.
I'm sure there was a plot, but I can't remember any details about it.