Which spell in Dota 2 has the largest Area of Effect?

There are a few contenders, but tied for first are Broodmother's "Spin Web" and Phantom Lancer's "Doppleganger" abilities, each with a radius of 900 units.

When looking at abilities that have offensive capability, the largest area of effect ability goes to Disrupter's "Static Storm", using a level 25 talent to increase the radius from 500 units to 775 units.

Without needing to level or improve the ability with talents or items, the largest offensive area of effect ability goes to the Techies' "Focused Detonation", with a radius of 700 units.

Broodmother's "Spin Web"

Broodmother's "Spin Web" ability creates a web that covers a radius of 900 units. The ability does not provide any offensive ability; that is, the Broodmother has increased movement speed when walking on a web, but they do not damage enemies.

"Spins a large web that grants Broodmother a passive movement speed increase, gives free movement, and boosts regeneration while in its vicinity. Spin Web charges restore every 40 seconds. Spin Web can be cast from anywhere as long as the new web touches an existing web. Webs never expire, and can be manually destroyed. When the maximum limit of webs is exceeded, the oldest web disappears."

- Description of Broodmother's "Spin Web" abiliy @ The DOTA2 Wiki

Phantom Lancer's "Doppleganger"

Phantom Lancer's "Doppleganger" ability allows both the player and any nearby illusions to teleport to another location. The initial area of effect comes in targetting the nearby illusions, and covers a radius of 900 units surrounding the player. All illusions reappear within an area of effect radius of 325 units.

"Phantom Lancer briefly vanishes from the battlefield. After 1 second, Phantom Lancer and any of his nearby illusions reappear at a random position within the targeted location, along with two additional doppelgangers. Extends duration of all illusions. The two added doppelgangers have different properties: one takes normal damage and deals none, while the other takes 500% bonus damage and deals 80% less damage."

- Description of Phantom Lancer's "Doppleganger" ability @ The DOTA2 Wiki

Disrupter's "Static Storm"

When looking at damaging effects, Disrupter's "Static Storm" comes in as the largest area of effect, with a radius of 775 units. This requires a level 25 talent that increases the radius up from 500 units.

"Creates a damaging static storm that also silences all enemy units in the area for the duration. The damage starts off weak, but increases in power over the duration."

- Description of Disrupter's "Static Storm" ability @ The DOTA2 Wiki

The Techies' Focused Detonate

When looking at damaging effects without the use of talents, the Techies' "Focused Detonate" ability comes in as the largest area of effect, with a radius of 700 units. The actual effect is that it triggers the detonation of mines, which need to be pre-placed by the player to damage enemies.

"Detonate all remote mines in the target area."

- Description of the Techies' "Focused Detonate" ability @ The DOTA2 Wiki