First off, it's important to define what inflation actually is. Inflation is the result of a currency (generally, in most localities, the state has a monopoly, or near monopoly on what is called "legal tender," -- i.e. the standard currency) losing its value per given unit thereof. Thus: if one dollar in the year 1 buys 1 pizza, and then, by the year 2, it only buys 1/2 of a pizza, we may say that the rate of inflation has been 100% per year, ceteris paribus.

Now, in the ancient world, this was no different. However, unlike today, in which our governments and economies largely run on so-called "fiat" currency (in other words, currency which is not backed by any physical standard, but rather simply by the credit and good-will of the government that supports it), in the ancient world, most coins were valuable in and of themselves. A Roman denarius, for example, had a certain amount of silver in it when it was first struck (weighing about 7 grams per coin at its first casting around 270 BC). Other coins had similar measures of "intrinsic" value.

However, as was very common, it was the government mints responsible for weighing out and striking these coins. As time went on, as you can imagine, there was a constant push for the rulers to keep the coins at their trading value, but hang on to as much of the physical silver as possible, either striking the coins with cheaper metals or just making hollow, lighter versions. This would allow them to have the coins, which were still worth the same amount per coin on the market when they were struck, AND have the extra silver/gold too. Eventually, the market would figure out that the coins were worth a lot less, and adjust prices accordingly, but the government/minters/banks would have access to this knowledge first, and would be able to trade with these coins before the prices adjusted, this converting the lower value coins into higher-value goods.

This form of inflationary practice is more commonly called "debasement," and the impetus to do it was utterly irresistible by governments ancient and medieval throughout history. Around 211 BC, for example, the denarius contained an average of 4.5 grams of silver. By the time of Nero, ~50 AD, it contained about 3. By the time of Gallienus, ~240 AD, it had been replaced by a cheap bronze coin with a thin silver wash. Remember that the theoretical VALUE was still the same, this was still a "dollar bill," but its actual worth was much less -- it was being debased. It was being inflated. At first, you could buy a whole pizza with one of these. Then just half. Then only a slice.

This cycle was repeated quite often throughout history, because the people in charge of making the currency ALWAYS had an interested in debasing it and inflating it. It is essentially no different from legal counterfeiting, with all the benefits that come from that. Many other examples exist throughout history: the dinar, descended from the denarius, in Medieval Spain went from sixty-five grains of gold to twelve grains in 200 years. In France, the livre tournois went from 98 grans of silver to 18 in four-hundred years. Both coins were eventually debased so far that they were dropped and replaced by "new" versions. This may sound familiar to anyone who is up to date on the history of the currencies of Zimbabwe or Venezuela, among other examples.

Inflation of currencies has been an endemic and ongoing problem for all of human history. It was definitely an issue in the ancient days, as well, although it could be combated to some degree by the barter system.

Sources:

• Groseclose, Elgin, Money and Man (New York: Frederick Ungar, 1961)

• The New Deal in Old Rome, HJ Haskell, Alfred K Knoff New York 1939

Wartime inflation was certainly common. In Aristophanes' 'Frogs' (link here) there is a joke where Charon (the ferryman to the underworld) demands twice his usual price to take Dionysus down to Hades - generally seen as a comment on wartime inflation during the Peloponnesian War during which the play was performed.

What's fun about the ancient Romans is that they just didn't get inflation. When their coin money started to be worth significantly less than it had in previous centuries, it came at a time when the silver content of the denarii had been reduced from nearly 100% to a fraction of that number. Emperors assumed that this was the issue at hand, so they simply ordered tons of new, properly silver-laden coins to be minted...which obviously made things worse considering that they never managed to take the old "lesser" coins OUT of the market.

This is all explained as sort of a background force during the middle-end parts of Mike Duncan's fabulous History of Rome podcast.

There is the Great Cowrie Inflation. Cowries had been in used in some parts of West Africa since the eleventh century and very probably earlier. During the slave trade, huge amounts of Maldivian cowries were introduced into Africa by the Portuguese further consolidating its widespread acceptance. German and French firms however began in the late 1840s and 1850s, to import very large cargoes of cowries from Zanzibar, (some 27,000 tons in 30 years). These were larger and heavier than the Maldives cowries, and therefore of lower value per hundred weight; they were, however, much more abundant and cheaper. The effect was disastrous. The "head" of 2,000 cowries, which had been worth about 4/6 to 5/6 at the beginning of the century, dropped to 1/6 by 1859, (when for a time none would accept cowries at all) and continued to fall. In 1904, when the damage was complete, the import of cowries was prohibited.

• The Cowrie Currencies of West Africa
• The Shell Money of the Slave Trade

Edit: Since this was removed earlier without any reason given, I'll assume it got caught in the spam filter.

There's a lot of theory but not a lot of concrete examples.

The first thing you need to appreciate is that in the days of yore coinage wasn't always the main mode of trade. Serfs paid rent in crops, not coin. Barter was common enough. It's not unusual for coinage to be the method of trade between nobles, and / or governments.

Second, there was much more need to guarantee the value of coinage. Skilled labor wasn't valued as it is today, and the concept of GDP when Ancient Athenians battled Ancient Spartans and the Assyrians were a thing, didn't exactly exist. For a coin to be worth anything it had to be inherently worth something- it needed a weight in some sort of metal. Fiat currency didn't typically happen on a very large scale. 300 years ago it was important for European powers to put their countries on gold standards to compete on an international level. 3000 years ago states had to standardize the value of their coinage so that anyone would ever want it. Even if a state folded, a mint changed the weights of gold to, say, copper, or what have you, that coin would still carry inherent value.

Inflation exists in an economy kind of like how there's always low and high pressure fronts somewhere in the global weather system. Because a high demand item is always shifting in value in an economy, the currency used to buy it then also changes in value. The issue is that we simply do not have detailed records of this happening. Accounting is, by some measures, a modern discipline.

Inflation is a relatively modern phenomenon, so I don't know if the ancient world experienced what we would call 'inflation' (a gradual controlled increase of prices).

There were economic crises in the Ancient world though. One example of a crisis (which isn't inflation) happened in 33 AD in Rome, under Emperor Tiberius

A severe reduction in governmental building programs following a period of very high investment created an economic condition of sharply reduced liquidity. In an economy where most construction labor is slave, loss of government contracts put severe pressure on slaveowners who had been contractors for government projects. The investment hiatus was then followed by a separate government decision to reduce interest rates in Rome below their natural rate and to require immediate renegotiation of existing loans down to this legal maximum. Lenders then moved their money to outside uses. The resulting panic was relieved by sharp, heavy infusions of government cash into the economy. - The Financial Crisis of A.D. 33: A Keynesian Depression?, Journal of Economic History 50, no. 3 (1990), p. 656

(citation comes from /u/ampseph who along with /u/tiako knows a ton more about the Roman economy)

Barbara Levick alleges that Tiberius responded to this financial crisis by confiscating (wealth-creating) assets in order to then use that money to fuel the economy. ("Tiberius the Politician" - Barbara Levick)

It should be noted that the Roman economy was (obviously) comletely different from moder economies. In a modern (capitalist) economy, responding to a lack of private investments by confiscating assets from private owners would be disastrous. If you google the Rome 33 AD crisis a bit, you'll run into some articles (like this one) which try to draw parallels between the Roman economic crisis and a modern-day economic crisis, which of course is completely ridiculous.