Historically, there have been periods of inflation in hard metals when supply outstripped demand, but not to the degree that is experienced with faith-based paper currency. Spain suffered a terrible bout of inflation due to imports from the Americas, for instance; this contributed to the downfall of a once-great empire.
Over longer periods of time, the technology for gold or silver tends to stay at par with the technology for producing other goods.
In an AnCap society, I'm going to assume at least a "free banking" rule where banks are not protected by law against bank runs, and therefore must have much higher reserve standards than are the norm today. An educated populace would demand - and get - 100% reserve banking. In a 100% reserve banking system, every transaction would be backed 100% by gold, silver, or some other acceptable medium of exchange. This would increase demand for gold and other precious metals. I'd expect a fair number of transactions using actual physical gold, for a variety of reasons: the joy of handling such beautiful coins, a desire to keep bankers honest, a preference for anonymity, and/or to avoid the fees of 100% reserve banks. ( A 100% reserve bank cannot possibly offer "free checking." )
People seldom think about how much pocket change they accumulate over long periods of time. I would guess anywhere between 5 and 10 dollars worth of change per week is typical. In a society where coins are preferred to paper, the amounts might be considerably larger.
If you haven't already done so, you ought to purchase some gold and/or silver. Compare these coins to the cheap junk now distributed, and you'll be amazed.
As I post this, the current price of gold is $44.56 per gram; little Babette made off with a fair bit of change.
100% reserve banking, zero money-from-thin-air, and no taxes would encourage savings - and most people, I suspect, would have substantial savings of tens or hundreds of thousands of grams of gold.