When will it end?

The following expressions are written so that they require an infinite number of terms.  Can you rewrite either of them in a closed form?

1.   \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}

2.   1+\displaystyle\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}

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