$\begin{array}{l} \sqrt {\sqrt 5 - \sqrt {3 - \sqrt {29 - 12\sqrt 5 } } } \\ = \sqrt {\sqrt 5 - \sqrt {3 - \sqrt {20 + 9 - 2.3.2\sqrt 5 } } } \\ = \sqrt {\sqrt 5 - \sqrt {3 - \sqrt {{{\left( {2\sqrt 5 - 3} \right)}^2}} } } \\ = \sqrt {\sqrt 5 - \sqrt {3 - 2\sqrt {5 + 3} } } \\ = \sqrt {\sqrt 5 - \sqrt {6 - 2\sqrt 5 } } \\ = \sqrt {\sqrt 5 - \sqrt {{{\left( {\sqrt 5 - 1} \right)}^2}} } \\ = \sqrt {\sqrt 5 - \sqrt 5 + 1} = 1 \end{array}$