Intuitive explanation of the tower property of conditional expectation I understand how to define conditional expectation and how to prove that it exists.

Further, I think I understand what conditional expectation means intuitively. I can also prove the tower property, that is if $X$ and $Y$ are random variables (or $Y$ a $\sigma$-field) then we have that

$$\mathbb E[X] = \mathbb{E}[\mathbb E [X | Y]].$$

My question is: What is the intuitive meaning of this? It seems quite puzzling to me.

(I could find similar questions but not this one.)
First, recall that in $E[X|Y]$ we are taking the expectation with respect to $X$, and so it can be written as $E[X|Y]=E_X[X|Y]=g(Y)$ . Because it's a funciton of $Y$, it's a random variable, and hence we can take its expectation (with respect to $Y$ now). So the double expectation should be read as $E_Y[E_X[X|Y]]$.

About the intuitive meaning, there are several approaches. I like to think of the expectation as a kind of predictor/guess (indeed, it's the predictor that minimizes the mean squared error). 

Suppose for example that $X, Y$ are two (positively) correlated variables, say the weigth and height of persons from a given population. The expectation of the weight $E(X)$ would be my best guess of the weight of a unknown person: I'd bet for this value, if not given more data (my uninformed bet is constant). Instead, if I know the height, I'd bet for $E(X | Y)$ : that means that for different persons I'd bet a diferent value, and my informed bet would not be constant: sometimes I'd bet more that the "uninformed bet" $E(X)$ (for tall persons) , sometime less. The natural question arises, can I say something about my informed bet in average? Well, the tower property answers: In average, you'll bet the same.
The end result is the same no matter how you slice it.

Think of the inner term of the RHS : E[E[X|Y]] as integrating over a specific direction, then the outer expectation integrates over the remaining directions.  This obviously skips important details but that's how I intuitively rationalize it.