Memories are precious things. Where would we be without memories? Who would we be without memories? Mages of the certain world can use them to create wonders.
The way it works is such: mages take a memory and intensify it, and throw it into the world with such force that the world recalls this event.
Recalled by the world any memory becomes powerful. Channelled by the mage into a narrow brief focus it becomes a useful spell. The memory of successful recovery from an illness becomes a fast recovery from an illness, the memory of scathing pain becomes an actual scathing pain, the memory of a lit bonfire sets the wood aflame in the same way as mortal's memory of their old humiliation can briefly flare their ears red again.
The world itself is non-being, it lives only in the present, knows no past, dreams of no future. The rare memories of the world are fossil detritus, stuck in random mountains and seas, discarded without care, valuable beyond measure. The world is ever-young because, without memories, it isn't aware of its passage through time. But every time a mage makes magic and throws a memory back at the world, the world pauses a bit, and learns a tiny bit, and becomes slightly older. Few mortals give thoughts about this and certainly whatever is happening is not going to affect their own lifetimes.
Many natural processes become wondrous when intensified in a such way. Even if mages cannot throw fireballs they can create burns, sent lightnings, weave rain, open wounds, speed up a natural recovery, and tilt chances in many things in their favour. Animal memories are brief, fragile and much weaker but a skilful enough mage can take a bird's memory of flying and fly themselves.
It would seem to be easy for a mage to take a bird's memory of flying, make somebody fly and harvest new memory then, in an endless loop of taking and using, using and taking, but the magic doesn't work this way. Memories caused by magic cannot be stolen and reused: the world imprinted them with all its brute force and no mage is powerful enough to take that until the bearer is broken down to something the world no longer recognizes.
Soul-healing is also an exception: it requires souls to work correctly, and its effect cannot be replicated by the mage. Such memories are still valuable but at the best they give a powerful natural recovery and good chances of cure: nothing fast, nothing convenient, nothing miraculous, nothing fully guaranteed.
Memories of monsters are extremely precious things because in this world monsters can naturally do what world itself (mostly) cannot naturally do; monsters are the remnants and seeds of otherwise truly impossible realities. Their abilities of turning people into stone, becoming invisible, breathing fire are sought by mages everywhere and many mages would give a half of their lives to pillage the memories of ancient crocodiles if they had a chance.
Unlike anybody's else, mage's own memories are never expended and this is why that world's mages are the craziest, the most inquisitive, most unfettered travellers you'd going to meet. They need to taste anything, touch anything, get into the most difficult and exotic locations, have friends and enemies and unhappy loves, and have their dreams crushed so they can do the same to others, and have their hope restored so they can do the same to others too. Some mages are just taking the life as it goes, and some are trying to engineer their lives into desired memories but the one way or another they'd sneer at the idea that the magic might be hidden in runes and books if they knew about such ways.
Burnout, the jadedness, the inability to feel and be surprised anymore is their only weakness as sooner or later the vast majority of mages hits this state. After that they either go to extremes of pain and pleasure just to feel anything new or settle down and go through their lives with what they managed to acquire. Alzheimer's is their professional disease and the biggest fear aside of other mages.
Rest of people are not so fortunate and do not retain memories that are cast into the world to create magic. Children's memories are the easiest to take - this is why we remember so little of our childhoods - but many of adult's memories, even the important ones, can be taken without permission as well - mages are not called "memory-thieves" for nothing. Just like memories caused by magic, their own memories are exceptions and can be taken by other mages only after all inner defences are broken or the mage is no longer recognizable by the world on physical, psychical and spiritual level: this is why most mages prefer to keep their enemies alive and torture them for power over the years.
Just like with souls there are ways to keep the memories in objects, to attach them to mementoes just like the world creates its memory-detritus; some mages go into business and make a career out of trading, creating, crafting on demand or stealing certain mementoes. Unlike souls, though, memories go stale in a while, least powerful memories first and most powerful (such as of monsters) after centuries. The only memories that aren't fading are the memories of world itself, but very few and far between, they are true rarity to find. Luckily, unlike monsters, the humans are renewable resource in this respect and there are always new memories on the market.
Aside of soul healing the only thing that mages can't affect it is time - they can speed or slow the perception of time, but the time itself is domain of the dead and only dead know its truth.
Inspired by several things
1) I wanted to see how Sensate-based magic might look like: Sensates believe that the universe can be understood only through experiencing it and hunt various experiences. In the 'Planescape' Fraternity of Order can use loopholes in reality for power because they believe in reality running by rules (with loopholes), so why Sensates cannot power up by using their own or others experience?
2) Running a theme "humans are renewable resource" I wanted to see if other non-corporeal parts of humans can be utilized aside of their souls for healing. Monsters and Manuals's idea of memory as of actual thing you can go into certainly helped here.
3) I also wanted to have buff mages that climb crazy mountains, do rapid-water slaloms, go into fights and eat strange foods all the time.
This also explains senile mages (and cenobite mages, and torturer mages) quite nicely.
As all inventory-based systems is probably unplayable in reality because its direct implementation would be really clumsy, with spreadsheets and micro-management of resources all the time.
So I think the mechanical system should be, in fact, the opposite of this: something fluid enough to simulate "I just have it in my bag maybe" effect of having or not having a particular resource on hand, presuming that any decent mage goes around with a stack of various stored effects. The system should take into consideration the skill of the mage and rarity of certain effects too. Some effects, 'their own memories' should be basically at-will abilities, and therefore either quite weak or carefully balanced with level, neither of two approaches is entirely elegant. This leaves truly powerful mementoes (such as memories of monsters, soul-healing and of the world) to keep track on.