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 $|x|\to\infty$ together with their derivatives go to zero faster than any power $1/|x|^n$. The "intricateness" of this space stems from the many operations you can perform in it and from the fact that these operations are intertwined with each other in miraculous ways. $$ $$ Responding to a comment: You have (a) ordinary multiplication and convolution, (b) "multiplication" with arbitrary polynomials $p(x)$ and operations $p(D)$, (c) multiplication with functions of the form $x\mapsto e^{iax}$ and the translation operator $T_a: f(\cdot)\mapsto f(\cdot-a)$ and (d) scaling of the variable $x$ resp. $\xi$. The Fourier transform $\Phi$ interchanges in each of these three cases the respective operations; and at heart of it all is Gauss' normal distribution $x\mapsto {1\over 2\pi}\int e^{-x^2/2} dx$ which stays fixed under $\Phi$.

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 $|x|\to\infty$ together with their derivatives go to zero faster than any power $1/|x|^n$. The "intricateness" of this space stems from the many operations you can perform in it and from the fact that these operations are intertwined with each other in miraculous ways. $$ $$ Responding to a comment: You have (a) ordinary multiplication and convolution, (b) "multiplication" with arbitrary polynomials $p(x)$ and operations $p(D)$, (c) multiplication with functions of the form $x\mapsto e^{iax}$ and the translation operator $T_a: f(\cdot)\mapsto f(\cdot-a)$ and (d) scaling of the variable $x$ resp. $\xi$. The Fourier transform $\Phi$ interchanges in each of these three cases the respective operations; and at heart of it all is Gauss' normal distribution $x\mapsto {1\over \sqrt{2\pi}}\int e^{-x^2/2} dx$ which stays fixed under $\Phi$. And, last not least, there is a scalar product which is preserved by $\Phi$.

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.

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 $|x|\to\infty$ together with their derivatives go to zero faster than any power $1/|x|^n$. The "intricateness" of this space stems from the many operations you can perform in it and from the fact that these operations are intertwined with each other in miraculous ways. $$ $$ Responding to a comment: You have (a) ordinary multiplication and convolution, (b) "multiplication" with arbitrary polynomials $p(x)$ and operations $p(D)$, (c) multiplication with functions of the form $x\mapsto e^{iax}$ and the translation operator $T_a: f(\cdot)\mapsto f(\cdot-a)$ and (d) scaling of the variable $x$ resp. $\xi$. The Fourier transform $\Phi$ interchanges in each of these three cases the respective operations; and at heart of it all is Gauss' normal distribution $x\mapsto {1\over 2\pi}\int e^{-x^2/2} dx$ which stays fixed under $\Phi$.