What, approximately, is the earliest possible date that the ballpoint pen could plausibly have been invented?
Ballpoints have existed since at least 1888, and was when the patent was filed, but not made until 1938. Reservoir pens had existed longer, all the way back to the 10th Century, and the Fountain was created in 1827. So, 1888 is the earliest, assuming someone experiments with the design.
But would it have been technologically feasible to make ballpoints before 1888, assuming someone had come up with the idea and been interested in exploiting it?