I'm currently in Melbourne, a coffee-lover's paradise. As such, I'm quite keen to try the infamous 'cat poo' coffee (Kopi Luwak) in Melbourne, given that the skill of a Melbourne barista is likely to surpass mine, and should be able to brew a good cup.
To be clear - I don't want to buy the coffee grains or bags of it, I want to buy a cup of it to drink.
The excreted deposits from the palm civet, after eating:
I'm not sure that any cafés will have this on the menu to be honest. What you could try doing is purchasing a bag of the beans and then taking it to a good barista and ask them to brew it for you. No guarantees that will work, but it might be worth a try. Particularly if you catch them when they're not too busy.
If you are on a quest to taste some great coffee while in Melbourne, you may be interested in this recent article on the most expensive cup of coffee in Melbourne (Jun 21, 2013). Price tag: $30.
