By typing alone, the answer is obviously no; the biggest weakness a Pokémon's typing can give it is 4x.
But there are some abilities that give Pokémon extra weaknesses, which is what I looked for. After looking through all the abilities that add weaknesses (and all the Pokémon that can have those abilities), the closest a Pokémon can come to an 8x weakness, without type manipulation, is the Paras line, who, in addition to having a natural 4x weakness to fire, can also have the ability Dry Skin, which increases fire damage by 1.25x, resulting in a 5x weakness to fire.
However, a Pokémon affected by Forest's Curse could end with an 8x weakness to various types. The total list of Pokémon who, when affected by Forest's Curse, have an 8x weakness:
FIRE
Alolan Sandshrew / Sandslash
Forretress
Scizor / Mega Scizor
Wormadam (Trash Cloak)
Escavalier
Durant
Genesect
ICE
Gligar / Gliscor
Landorus (both forms)
Vibrava / Flygon
Gible / Gabite / Garchomp / Mega Garchomp
Zygarde (all forms)
Dragonite
Altaria
Salamence / Mega Salamence
Rayquaza / Mega Rayquaza
Noibat / Noivern
FLYING
Heracross / Mega Heracross
Buzzwole
Pheromosa
BUG
Inkay / Malamar
Unbound Hoopa
(credits to people in the comments to pointing out Forest's Curse)