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Addendum: actually there is a chain of subfields between $K((x,y))$ and $K((x))((y))$ with cardinality c. For instance, for any $\lambda>1$ we may consider the subset $R_\lambda$ of $K((x))((y))$ of all Laurent series $\sum_k c_k(x)y^k$ with $c_k\in K((x))$ satisfying $$\inf_k \lambda ^k {-k} \mathrm{ord}(c_k) > -\infty.$$ It's easy to check that it's a subfield of $K((x))((y))$, containing $K[[x,y]]$.

Moreover, since in place of $\lambda^k$ we can use functions $\mathbb{N}\to\mathbb{N}$ with arbitrarily large growth, one can also show that the subfields between $K((x,y))$ and $K((x))((y))$ have uncountable cofinality also.

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Addendum: actually there is an uncountable a chain of subfields betewwn between $K((x,y))$ and $K((x))((y))$. K((x))((y))$with cardinality c. For instance, for any$\lambda>0$\lambda>1$ we may consider the subset $R_\lambda$ of $K((x))((y))$ of all Laurent series $\sum_k c_k(x)y^k$ with $c_k\in K((x))$ satisfying $$\inf_k \mathrm{ord}(c_k) e^{-\lambda lambda ^k } \mathrm{ord}(c_k) > -\infty.$$ It's easy to check that it's a subfield of $K((x))((y))$, containing $K[[x,y]]$.

Moreover, since in place of $\lambda^k$ we can use functions $\mathbb{N}\to\mathbb{N}$ with arbitrarily large growth, one can also show that the subfields between $K((x,y))$ and $K((x))((y))$ have uncountable cofinality also.

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Addendum: actually there is an uncountable chain of subfields betewwn $K((x,y))$ and $K((x))((y))$. For instance, for any $\lambda>0$ we may consider the subset $R_\lambda$ of $K((x))((y))$ of all Laurent series $\sum_k c_k(x)y^k$ with $c_k\in K((x))$ satisfying $$\inf_k \mathrm{ord}(c_k) e^{-\lambda k } > -\infty.$$ It's easy to check that it's a subfield of $K((x))((y))$, containing $K[[x,y]]$.