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I have voted for ${\mathbb N}$; but let me nevertheless propose an object living in the analytical realm, namely the Schwartz space ${\cal S}$ of infinitely differentiable functions $f:{\mathbb R}\to{\mathbb C}$ that for $|x|\to\infty$ go to zero faster than any power $1/|x|^n$. The "intricateness" of this space comes from the many operations you can perform in it and from the fact that these operations are intertwined with each other in miraculous ways.