Well Remington "officially" says 1908 to 1929 but that's not entirely true. It's actually 1907 they made less then 100 (actually 52 of them) but they did make them and as late as 1931 where they made a couple hundred. The actually factory records do show this but I guess because the numbers are so low??
According to Remington, it was made in 1913.
1909
1910
1920
A fortune since it would have to been made by hand as a prototype nearly 10 years in advance of production.
Early 1914
I can tell you it was made in 1928, that's about it. If the barrel is original you can get the month/year from the date code stamp. See related links. Also the Remington model 10 was made by Remington from 1911-1929
1928, will be date stamped on the barrel check the link to the Remington Society of America.
Made in 1908
The Remington Repeating Shotgun was introduced in 1907. The name was changed to the Model 10 in 1911. In excellent condition, it has a blue book value around $300.
Try this link from Remington. http://www.remington.com/library/history/firearm_models/shotguns/model_10.asp
The Remington nylon 10s were made (1962-1964) before serial numbers were required (1968)and I think all of them were made without a SN.