Type effectiveness is a very narrow classification. The only things that fall under that description are the actual type chart multipliers (as possibly modified by Freeze-Dry and Flying Press's special mechanics), plus Tar Shot (creates an additional x2 multiplier against Fire moves) and Tera Shell (replaces whatever the type effectiveness would be, with 0.5 if Terapagos is at full health). STAB is not treated as a type effectiveness multiplier, weather is not treated as a type effectiveness multiplier, and even Dry Skin or Fluffy are not treated as type effectiveness multipliers.
Counting only things that are actually type multipliers, the best you can do in the current games is x16, achieved by using Forest's Curse (Trevenant's signature move) against a Pokemon like Scizor or Snom to give it a third type, Tar Shot (Coalossal's signature move) to give it an additional Fire weakness separate from any type, then hit it with a fire move. The highest type multiplier that has ever been legally achievable in any game is x32, for which you have to play in an Inverse Battle (most easily done in gen 6), have Smeargle Skill Swap its own ability to Normalize (Delcatty's signature ability), have the opponent send out an Aggron, then use Trick-or-Treat against the Aggron and finally hit it with Flying Press, which thus counts both Normal and Flying type matchups on the Inverse Battle type chart against Aggron's Steel/Rock/Ghost typing. The programmers designed this field to hold values as large as x64, but there's no way to reach that limit.
Other effects generally go into one of three places: move power multipliers, stat multipliers, and damage multipliers.
Move power multipliers include things like plates, other type-boost items, terrain boosts, Helping Hand, some ability boosts like Technician or Analytic, Dry Skin (against fire moves, it works as a move power multiplier on the opponent's move), and Rollout's escalating effect that can count up to x16 or even x32. Enumerating which of these multipliers can possibly coexist is complicated, variable depending on the game, and also not the point of the question.
Stat multipliers include the basic stat stages; items such as Light Ball, Thick Club, and Eviolite; a different set of abilities like Overgrow or Solar Power; Thick Fat (when this ability applies, it's actually implemented as a multiplier that reduces the opponent's attacking stat), and Flower Gift in games where that ability is usable. A Marowak with +6 Attack, Thick Club, its ability swapped to Pure Power, and a Cherrim teammate for Flower Gift in sun will have an Attack stat multiplier of x24. If you're playing gen 6 so that Triples exists, and give it two Flower Gift teammates at once, you can get to x36 that way.
Damage multipliers include things at the end of the calculation after you've already combined the other numbers. This is where STAB and weather multipliers go, and they also include screens, Life Orb, yet another set of abilities like Sniper or Tinted Lens, and the opponent's Solid Rock or Fluffy.
