Yes, lots of folks have found quite a few coins pertaining to the early colonies. The Red Book of US Coins now has a small section in it of coins used in the Americas before the constitution was put in place.
I have dug over the years Spanish silver coins in 1/2, 1, 2, 4 and 8 Reale denominations. They were legal tender here until about 1853, when congress made use of them illegal. Spanish coins are some of the most common pre 1787 coins to find. Sometimes you can find a 1/2 blanca, a small denomination Spanish coin or some in the maravedis series. They were made out of hammered copper.
British halfpennies and farthings are much more common than even the Spanish coins I listed. Lots of home places had a halfpenny or two and lost them. Most are from the reigns of William and Mary, one of the oldest found, George I, II and III.
There were some silver coins (British) that occasionally turn up but they are much harder to get. Also British Gold Guineas sometimes show up.
An occasional French colonial issue is discovered, but rarely. The same with Lion Daadler from the Netherlands. I only know of a few ever found.
On extremely old sites people have found coins from the reign of Elizabeth I, and sometimes James I. Some other items it is possible to find are the Machin mills coppers, I don't think I have ever found one, Woods Hibernia coppers and Rose Americana coppers. All nice stuff, and hard to get. The Massachusetts Pine and Oak tree shillings and sixpence coins are found in the northeast, and sometimes as far south as Virginia. I wish I could have found one of them.
So, yes, there are a lot of different types you can theoretically find, but you have to get on a very old site to do it.
