Fantasia

[Enkidoh]

Getting back to the original poster's initial question, yes, it is (or rather, was) possible to get up to 4 free bottles of Fantasia, but the ways for doing so are either no longer available or extremely difficult now.

A free bottle of Fantasia was included with the physical Collector's Editions of A Realm Reborn and Heavensward (but not Stormblood). The physical Collector's Editions were only available in pre-order however and are thus are almost impossible to find now (although I think you can get these items through the 'Digital Upgrade Service' that SE offered that allows you to gain the included items with the CEs on a 'standard' copy of the game, but this costs extra).
A third bottle was given free to players who created a character in the original 1.0 version of FFXIV several years ago now (it was sent through the Moogle Delivery System).
Finally a fourth bottle was an early Veteran Reward for subscribing to the game for a certain length of time (2 months if I remember rightly). This was removed when SE overhauled the Veteran Rewards system when SB launched, and temporarily made the Fantasia a quest reward for completing the original '2.0' main scenario in the game accordingly (in other words, the main scenario quest 'The Ultimate Weapon'), but as I understand it that has since been removed sadly, with no other way to acquire it I'm afraid.

Either way, fantasia is/has been offered to players for free, but these incidences are the exception rather than the rule unfortunately, so purchasing Fantasia through the Mogstation is the only real way to acquire more. As there are more than a few 'Fantasia addicts' out there, this may not necessarily be a bad thing.