According to Memory Alpha, Latinum cannot be replicated. What is the reason for that?
According to several sources, including the TNG novel "Balance of Power", latinum is described as being so dense and complex that the replicators are unable to restructure matter into a form that duplicates it. In its natural state, latinum is a liquid which is mined from certain types of nebulae. It is pressed into gold in order to make it more usable as a means of currency.
Of course, the explanations came later. Originally it was simply a story element created as a means of having non-credit monetary transactions in a universe where a replicator could create anything you want.
I think you're coming at this from the wrong direction. The question shouldn't be 'Why can't you replicate latinum', it should be "Why is latinum used for currency?"
The basis for currency is something that can't be easily created, can be easily transported, and is limited in quantity. Obviously, the Ferengi chose Latinum because it fit the criteria. It obviously isn't perfect - latinum, as BBlake pointed out, is a liquid. Liquids are horrible for currency - easy to lose portions of them, difficult to transport, hard to exactly measure, and hard to obtain (typically - you can't usually mine for liquids).
If there had been a solid that fit better, I'd bet anything the Ferengi would have used it instead. Instead, they go with latinum, pressed into small gold bars. The only good reason they could have for choosing latinum is that it isn't possible to make via replicator.
Ignoring for the moment the entire question of how a replicator would work in the real world (this is fiction after all), the problem with latinum not being replicable is non-trivial.
We have been told that replicators can't handle some kinds of radioactive materials - which makes a kind of sense since the material being replicated would changing continuously - except that's an argument for why some materials can't be transported. It would also mean that latinum is constantly decaying into something else - and that would make it a terrible form of currency.
The whole problem stems from a core problem with ST science - the writers don't seem to fully grasp what's an element, a chemical, a subatomic particle and so on (don't get me started on that horrifying stupid 'omega particle' rubbish introduced in Voyager - or worse, 'red matter' in the first new ST movie... A chemical way to create black holes? Really?).
Latinum seems to be an element (like gold). In fact, it kind of has to be by reduction: if it's a chemical, it's made of elements and we're back to elements (same argument for alloys or amalgams). That implies then that there's a non-radioactive element that's not replicable.
Thing is - no matter how complex (which can only mean 'large') an atom gets, it's going to operate on the principles of quantum mechanics which are predictable. Any process which can take energy and essentially squeeze it into electrons and quarks and then glue quarks together to make protons and neutrons is already going to have more than enough power and precision to make any atom, no matter how complex.
Sooooo... @neilfien's answer is basically dead on, and @BBlake's answer has the reason.
@Jeff's answer while technically correct is putting the cart before the horse and isn't so much answering the question as justifying it. :)
Latinum being radioactive would make a heckuva lot of sense. It would be the reason why Mourn's hair fell out from holding the latinum in his second stomach, and it would also be the reason why it is encased in gold.
Gold is a decent radiation shield, and since there is not a lot of latinum in each brick/bar/stick/slip compared to the amount of gold used, it's safe to say that it is sufficient enough to shield the radiation that the Latinum produces.
It would also explain why it can't be replicated, if replicators can't replicate certain radioactive materials (as someone above me posted).
I'm gonna take a shot at this. Remember that even the food and alcohol the replicators make are not exact to the real thing. It's a synthetic version. I would venture to say that seeing how difficult complex metals and organisms can be from raw material, it probably just can't produce anything over a certain technical rating. You can tell the replicator how make it, But it will only do so much thus the synthesised title. You could then scan to see if the latinum had these "cheat" atomic shortcuts and then know if its fake or real.
Answer on this question can be done by cross-search of items which can not be replicated. And it quite clear from my point of view that issue lies in low resolution of replicator. I guess latinum has molecular structure which contains some particle with low size. For example: It can be stable neutrino matter as it was showed in Solaris.