You know, there is a school of thought which says that a stockpile of nuclear weapons big enough to kill every living thing on the planet is big enough, and any extra are probably unnecessary expense. A nuclear deterrent only needs to be large enough to completely and totally annihilate any country that may attack you. The British nuclear arsenal is big enough for that. The US has about an order of magnitude more.
In order to be an effective deterrent, you need a stockpile large enough to survive a nuclear first strike, and penetrate the enemy's defenses.
"Big enough to kill everything on the planet" is quite arbitrary, if your enemy thinks they can take out most or all of your nukes before you can retaliate, then you have no deterrent at all.