What are some of the most difficult mathematical concepts for mathematicians to grasp?

[sumansuhag]

This is obviously subjective and argumentative, but hey. Some concepts that come to mind, in no particular order:

Spectral Sequences. "The subject of spectral sequences has a reputation for being difficult for the beginner." writes Tim Chow in "You Could Have Invented Spectral Sequences". He goes on to quote Whitehead: “The machinery of spectral sequences, stemming from the algebraic work of Lyndon and Koszul, seemed complicated and obscure to many topologists.”
Motives (in algebraic geometry). Not only are they hard to grasp, they haven't even been fully defined yet, at least not to everyone's satisfaction. Many of Grothendieck's formed or half-formed ideas are challenging for most ordinary mortals, like topoi and stacks.
Vertex Operator Algebras. Look up the definition on Wikipedia and you should get a sense for why it's not an easy thing to grasp.
Non-commutative spaces, as in Connes' non-commutative geometry.
Large cardinals. Most of the definitions are reasonably easy to come to terms with if you're familiar with the terminology, but to say anyone "grasps" a superstrong cardinal, for example, is kind of a stretch.

[vladlen-codes]

You've nailed several of the classics lol. I'd particularly agree about motives but we're still figuring out what they even "are" and large cardinals you can know the definition without really grasping what you're talking about.
Spectral sequences: might be more "intimidating machinery" than genuinely hard to grasp once you've seen enough examples but the learning curve is brutal.
I'd add: the étale topology (notoriously counterintuitive) and maybe the classification of finite simple groups are too vast for anyone to hold completely in their head.
The common thread seems to be concepts where either the intuition fundamentally lags behind or where the object is defined more by what we "want" it to be than by what we can actually construct.