What abilities have an additional effect

Off the top of my head:

These abilities activate when the Pokemon is in the first spot of the party:

Intimidate & Keen Eye: Decreases the chance of encountering lower-level wild Pokémon by 50%
Stench & White Smoke: Decreases the chances of encountering wild Pokémon (not just lower level) by 50%
Arena Trap & Illuminate: Increases the chance of encountering Pokémon to 200%.
No Guard - Increases the wild Pokémon encounter rate to 150%.
Static: Increases the chance of running into Electric type Pokemon by 50% (in areas that have electric types)
Magnet Pull: Increasess the chance of encountering wild Steel-type Pokémon by 50% (in areas that have them)
Synchronize: 50% chance that wild Pokémon encountered will have the same Nature as the Pokémon with Synchronize.
Cute Charm: Increases the chance to run into the opposite gender to 66.7%, regardless of normal gender ratios.
Compound Eyes: Increases the chance of wild Pokemon holding an item from 50%/5%/1% to 60%/20%/5%.
Swarm: Increases the chance of encountering wild Pokémon.
Suction Cups & Sticky Hold: Bites will occur more often while fishing.
Hustle, Pressure & Vital Spirit: Increases the chance to run into higher-level Pokemon.
Sand Veil: Decreases the wild Pokémon encounter rate by 50% in an area with a sandstorm
Snow Cloak: Decreases the wild Pokémon encounter rate by 50% in an area with a blizzard
Quick Feet - Decreases the wild Pokémon encounter rate.

Most if not all of the above abilities are still active even when the Pokemon is fainted.

These Abilities will activate regardless of their position in the party:

Flame Body / Magma Armour - Halves the number of steps needed to hatch eggs.
Pickup - 10% chance of creating a held item for itself, even if the Pokémon was not used in battle.
- Still active while the Pokémon is fainted
- Items get better the higher the level of the Pokemon
- Wont activate if the Pokemon is holding an item.
Honey Gather - Similar to Pickup, may collect Honey after a battle
- Wont activate if the Pokemon is holding an item.
- The chance increases the higher the level of the Pokemon. 5% at levels 1 and 10, and going up by 5% every ten levels, ending at a 50% chance from levels 91-100.

In generations past:

Immunity: If your Pokemon is poisoned, it won't take damage outside of battle.
- Theoretically still active in the current gen, however poison damage outside of battle has been removed.
Hyper Cutter: (Pokemon Emerald) Increased the range of Cut when used on tall grass. From Gen 4 onward you are unable to Cut tall grass
Lightning Rod: (Pokemon Emerald) Trainers registered with the PokéNav's Match Call function will call more often if a Pokémon with Lightning Rod leads the party.

Source for the ones I missed originally:

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