MTG Salvation

Annihilator is an extremely powerful mechanic that can really shut a person out of a game, especially as hits ramp up. Since you're likely to hit where the most damage is done, it can very quickly eat into a player's manabase, which can really cripple someone and put them out of the game essentially. I've built an Eldrazi deck, and had to make the choice between cutting Annihilator, cutting haste effects, or trimming down both, just because of how unfun it is.

That said, my vote actually goes to Ulamog, the Ceaseless Hunger. Both versions are extremely strong, and will be 'stronger' depending on what you are trying to accomplish, but my thoughts go like this:

• Ceaseless Hunger has a considerably more powerful 'on cast' ability, giving much better immediate value. It hits two targets (essentially 5 mana per 'removal', which is pretty good effectiveness in colorless), instead of one, and also exiles instead of destroys. One of the main reasons to run Ulamog is for a colorless removal option tied to a big beater, so better removal is strongly welcome.
• Ceaseless Hunger costs 1 less. The difference between 10 and 11 mana may actually not be that big of an issue when you plan to ramp up there anyways, but 1 extra mana is still an advantage.
• In terms of game ending threats, Inifinite Gyre's Annihilator is considerably more powerful of an ability since it impacts the board, so even if removed the value gained is good, while decks that run Ceaseless Hunger may not have ways to follow up on the mill should he get removed. However, exiling 20 cards is still a potent game closer, often taking only 3ish attacks to really end someone's game, despite any presence of chump blockers. Certain decks may have a considerably easier time dealing with Inifinite Gyre's Annihilator, and can just chump him for days (token decks). As such, Ceaseless Hunger's game closeout potential is still quite strong as well.
• Very niche, but there are some fun things you can do with Ceaseless Hunger's exiled cards, such as Oblivion Sower.
• In terms of 'fun', Ceaseless hunger generally offers more potential, and Infinite Gyre is considerably more detrimental to others' play experience.
• Combined with Haste, however, Infinite Gyre can be a much bigger surprise board-state wrecking tool, upping destruction to 5 permanents (cast+annihilator) for only 11 mana, barely more than 2 mana per destruction. If this is following a wrath effect, or some other situation where you catch someone with their pants down, even this one hit can be backbreaking. Ceaseless Hunger doesn't have quite that same impact on a hasted attack.

If I were building a deck, I would probably usually end up including Ceaseless Hunger before Infinite Gyre in most circumstances.