The type matrix for Flying Press is as such:
Super effective against: Normal, Fighting, Grass, Ice, Dark
Resisted by: Flying, Poison, Psychic, Electric, Fairy
The remaining types (Ground, Rock, Bug, Steel, Fire, Water, and Dragon) all give neutral damage. Note that in the case of Rock and Steel, the Fighting component comes up super effective but is cancelled out by the Flying component to produce the final result of neutral, while in the case of Bug, the opposite is true.
In particular, there is no single type on the chart where both Flying and Fighting are super effective against it, thus no way to get a 4x type multiplier against a single type, or 8x or 16x against a dual type. Of the super effective types, Normal, Ice, and Dark realize that weakness solely because of the Fighting component, while Fighting and Grass realize it solely because of the Flying component, with no overlap.
However, there are at least two ways that Flying Press can realize an 8x or greater multiplier:
1. Give the Pokemon a third type. Since the most any type can normally contribute is 2x to the multiplier, getting to 8 would require at least three of these multipliers stacked together, but no Pokemon has more than two types in its ground state. But for as long as Flying Press has existed as a move, so have the moves Forest's Curse and Trick-or-Treat, which can add types (but only one of those two moves can be in effect at a time). Fortunately, Forest's Curse adds the Grass type, which is one of the five types that gets hit super effectively on Flying Press's type matrix. So by using it on a Pokemon that already has two of the other four super effective types (Normal, Fighting, Ice, or Dark), the result is three types, all of which are weak to the move, and that does produce an 8x type multiplier. The other such move, Trick-or-Treat, normally just adds an immunity to the move and is of no help. But...
2. Play in an Inverse Battle...along with some other trickery. In addition to introducing the moves Flying Press, Forest's Curse, and Trick-or-Treat, X and Y also introduced the concept of an Inverse Battle, in which type matchups that would be super effective now become not-very-effective, and NVE moves plus type immunities now become super-effective. (Ability-based immunities, such as Levitate or Water Absorb, are still intact, and are the only ways to have immunities of any kind in inverse.)
Just as there are no types for which the Fighting and Flying matchups overlap as super effective, there aren't any that overlap as not very effective either, so an inverse battle would normally put you in the same position. But there are a couple extra tricks we can perform with move types, which wouldn't produce any payoff normally, but inverse battles do allow them to do so. Specifically, the type of the move can be changed by having a faster Pokemon use Electrify (in which case Flying Press will check the Electric and Flying matchups, not considering Fighting at all) or by using Skill Swap or Entrainment to change the Flying Press user's ability to Normalize (in which case it then checks the Normal and Flying matchups). In the Electrify case, where it's checking Electric and Flying, there is one type--Electric itself--that happens to resist both of those types normally, and is thus weak to both of them in inverse. Magnezone, Stunfisk, and Zekrom all combine Electric with one of the other types that's weak to Electrify Inverse Flying Press, and if you can get into position to actually hit them with such a thing, you really do get to achieve an 8x type multiplier against an opponent with only two types.
The Normalize variation goes even further. By considering Normal and Flying, not only is there a type that typically resists both of them, in fact there are two such types: Rock and Steel. This means that if you can hit Aggron with a Normalize Flying Press in an Inverse Battle, the result is a 16x type matchup. What's more, Trick-or-Treat can also be deployed here. Trick-or-Treat adds Ghost, which is normally immune to Normal (therefore weak to it in Inverse), but has no unusual interaction with Flying. Still, by hitting Aggron with Trick-or-Treat prior to the Flying Press, we can add a fifth distinct weakness, for an otherwise impossible 32x type multiplier. Outside of Inverse Battles, the best you can possibly do with any move (strictly in terms of the type multiplier, nothing else) is 16x in Sword and Shield, where you can use a Fire move on Bug/Ice + Forest's Curse + Tar Shot.
The trouble here is that Inverse Battles aren't always the easiest things to set up. XY have one house where you can participate in exactly one such battle per day, and your opponent uses a randomly selected team from among a bunch of predetermined sets, which aren't typically amenable to setting up the things you might want to do to test this. Omega Ruby and Alpha Sapphire are more helpful here, because the owner of a Secret Base can set their preferred battle type to Inverse, and select three custom Pokemon to be their team for it, then other players can find them and battle against them. Sun/Moon and Ultra Sun/Ultra Moon went back to slim pickings, as the only way Inverse Battles have ever been playable there was as the designated Special Battle format for just one ranked Battle Spot season per game. In the Ultra games' case, that stands alone as the only place where it has ever been possible to play an Inverse Double Battle, but those formats have long since expired and it is no longer possible to play them in those games. Meanwhile, Sword and Shield have not offered any way to play an Inverse Battle there, ever.