1. FFI - Started it all. Honestly, gtfo if you call yourself a FF fan and haven't at least attempted the game that started it all in one of its many verions.
2. FFIV - The first 'real good' FF game in my view. It introduced many aspects of the series (characterization, ATB) that would define it for years.
3. FFV - I could see the argument not to include V and instead include Tactics. You need to play at least one of these so you can see the true potential of the job system. But Tactics isn't on your poll so I went with FFV.
4. FFVI - The greatest video game ever and if you haven't played it then you are not a FF fan no matter what you say. Even if you hated it you cannot call yourself a fan of a series and not play its masterpiece.
5. FFVII - Mandatory FFVII inclusion. I don't think I even need to say why this should be there. Defined a generation, made RPGs mainstream, blah blah.
6. FFIX - Another one that is on the fence. FFIX is a great game and really returns to some of the charm of the 16 bit entries. It doesn't really do anything new or original but it's the definition of 'a FF for FF fans'.
7. FFX - I included FFX because it is important to also play one of the 'new school' FF games. FFX in many ways set the tone for the PS2 era and beyond. Perhaps in a bad way, but it was significant nonetheless.

Just saw above that you said limit it to 5. So obviously I would remove FFV and FFIX for my five:

1. FFI
2. FFIV
3. FFVI
4. FFVII
5. FFX