The cheapest "ancient coins" you can buy are mixed uncleaned Late Roman bronzes; these can usually be obtained for $5 to $10 each. They are sold unidentified, and will quite probably remain unidentified as they are likely to be too badly worn, corroded or damaged to identify properly. But they will be genuine 1600 to 1800-year-old Roman coins. Dirty Old Coins is a company that sells uncleaned and cleaned-but-unidentified ancient coins in bulk.
The more information is known about an ancient coin, then (usually) the more valuable it is. Take that same unidentified late Roman bronze you bought for $10, and be able to identify the emperor, and you've suddenly doubled the price. Double it again if the emperor's name is fully readable and the mintmark and reverse type is fully identifiable. Rarer emperors, types and mintmarks are going to be more expensive again. So there's a tradeoff: if you just want something really old in your collection and don't really care exactly what it is, then you can get one (or even a handful of them) really cheap. If you want an ancient coin that actually looks like a coin and not a small green rock, and about which something is actually known and knowable such as emperor, approximate date, mint-city, etc, then you'll have to pay a bit more.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis