To further complicate the already complicated Roman coinage situation I've talked about before, coins minted with the approval of the Imperial government weren't the only coins taking an active part in the Roman economy.
It shouldn't surprise you that there were counterfeits made for profit. The usual way of doing that is to make a blank out of a cheap metal, then wrap it in a layer of silver or gold foil and strike it. If you pull it off right, it should look like a legitimate precious metal coin until it takes some wear or someone makes a test cut to check its contents (something you see commonly on Classical Athenian tetradrachms by rarely see by the Imperial Roman period). This kind of forgery is called a fourrée, and you can see an example mimicking a denarius of Trajan (reigned 98-117 CE) above. These are hard to date, since while the government would normally depict the current emperor or if they were depicting a former one, make it clear that it's a posthumous thing, there's nothing stopping a forger from making a coin depicting whoever whenever. If anything, going a bit back might make the forgery more believable, since it's harder to tell if a forget got the style wrong if you don't normally see real examples (assuming you didn't just steal official dies, something that happened sometimes). The inscription makes reference to Trajan's Dacian campaign, so it couldn't be from earlier than 102 CE, but otherwise I wouldn't date this as anything narrower than "probably second century CE." This one was not made from official dies, as it contains typos that are otherwise unattested (like "SPR" rather than "SPQR" on the reverse), though there are rare cases of official coins having errors like that.
The next two coins are something more interesting to me, as there's a good argument to be made that they would never have fooled anyone into thinking they were official and never had any intention to do so. These things are usually called "barbarous radiates," though "barbarous" is used as a label for crude style rather than who made them, which was almost certainly Roman citizens. The usual thought1 about why these exist is that when Aurelian conquered the breakaway "Gallic Empire" in 274 CE, he closed the Trier mint. This meant that there was suddenly a lower supply of imperial coinage making it into northern Gaul and Britain. So the locals decided that if Rome wasn't going to give them the small change they needed to deal with the coin shortage they were experiencing, then they'd just solve the problem themselves. Thus, we have these crude, often rather small imitations of Roman coins. In the last picture, you can see my smallest example on top of an official coin that was already on the small end for an official issue. That little guy is really thin, too, and I think it weighed around 1 gram when I checked (I need to dig up that scale to confirm), less than half of even the lightest official issues. And the unofficial status of this coin was even more obvious if you were literate, as I'm pretty sure whoever carved that die wasn't, giving us some gibberish that looks like "IIIIVV" on it. These usually are imitations of Gallic coins, though sometimes you find imitations of a posthumous coin of Claudius II Gothicus, like the third coin in the pictures. Production of coins like these tapered off as more official mints were opened in Gaul and Britain.
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It's important to keep in mind that these are completely unmentioned in any literary source, so it's hard to know much for sure about them. There have been suggestions that these were made while the Gallic Empire was still active, and I've seen at least one person suggest they kept making these for centuries. It should be noted that we often find these in hoards with official Roman coins, and to my knowledge, we haven't seen them otherwise mixed in with much later coins, at least not without Roman coins also being there, so I definitely lean against that latter idea. What I'm presenting is the framework that makes the most sense to me on these things, but this is an area where the right archaeological find or the right scholarly analysis could end up completely changing how we see these things.