Trust
Is the most valuable currency. No matter the proposed value of an object, if you do not trust that you can change bank notes / coins / gold / platinum / diamonds / etc for something else, you will not trade.
One of the useful parts of currency is that you don't need to keep translating two chickens into 50 kilo's of apples. And 10 thousand apples make one cow. Etc.
So if your creatures have very good memories, what they probably need without active sensory input, as hearing and smell are passive, you need to remember where (your) stuff is.
If that is to much for you, use shells from certain snails for currency. Just got to make sure they are hard to forge.
To clear it up a bit. Every one has a running tab when you buy from one another. Or better yet, a very broad system of non-enumerated credits and debts.
So let's say I like to drink a beer and I go to the pub a few times a week to do so. The bartender knows me, and I have no easy way of buying each and every beer, so I have a tab running. Either he remembers, or notes it down. Now, sometimes the bartender will need to close the tab with me. Let's say he does so every month. 4 weeks * 5 evenings * 3 beers = 60 beers. Let's say that is worth 2 chickens. Or he clears the check ones every year. That might be worth a sheep. But if that sheep is worth 13 months of beer, I just have pay for a new month already.
Recommended reading: Debt: The First 5000 Years.
