User blog comment:DontDrinkH20/Some Explanations of Certain Large Cardinals/@comment-11227630-20181019090840/@comment-36540075-20181020200139

What exactly do you mean by "rich" in structure? I believe "almost Ramsey" cardinals might be very "rich" in structure if you mean that the size is quite nuanced; the least almost Ramsey cardinal is incredibly large yet only has cofinality \(\omega\) if it exists.