What is $\mathbb{E}\left[y \mathbb{E}\left[ y|x \right] \right]$?
My attempt:
$\mathbb{E}\left[y \mathbb{E}\left[ y|x \right] \right]$ = $\int\int y \mathbb{E}\left[y|X=x \right]f_{x,y}(x,y) dxdy $ = $\int \int y^2 \frac{1}{f_x(x)} \int f_{x,y}(x,y)dy f_{x,y}(x,y)dx dy $ = $\int \int y^2 f_{x,y}(x,y)dxdy$ = $\mathbb{E}\left[ y^2\right]$
Well, your attempt is not right and there are a lot of counter examples. For example, take $X=1$ a.s. and $Y\sim \mathcal N(0,1)$. Then $$ \Bbb E [Y^2]=1, \quad \Bbb E\bigl[Y \cdot \Bbb E[Y\mid X]\bigr] =\Bbb E\bigl[Y \cdot \Bbb E[Y]\bigr] = \Bbb E [Y]^2=0. $$
Without any further information on $X$ and $Y$ you can always get the following expression. $$ \Bbb E\bigl[Y \cdot \Bbb E[Y\mid X]\bigr] = \Bbb E\Bigl[ \Bbb E\bigl[Y \cdot \Bbb E[Y\mid X] \mid X \bigr] \Bigr] = \Bbb E\Bigl[ \Bbb E\bigl[Y \Bbb \mid X \bigr]\cdot \Bbb E\bigl[Y\mid X\bigr]\Bigr] = \Bbb E\Bigl[ \Bbb E\bigl[Y \Bbb \mid X \bigr]^2\Bigr]. $$