according to well known theorems, every valuation of a field $K$ can be extended to a valuation of a field extension $L$ of $K$ and this can be explicit in the case of finite extensions.
My question is how explicit this can be made in the case of an iterated Laurent series field: If we have $K=F((t_1))$ for a field $F$ and $L=K((t_2))=F((t_1))((t_2))$, how can the extensions of the $t_1$-adic valuation on $L$ look like? Is there something that can be said about the valuation on $L$ (archimedean? complete? uniformizer...)?