Square Root Inside Square Roots Inside Square Root!

Relevant wiki: Simplifying Expressions with Radicals - Intermediate

$\small{\sqrt{121}=11\\\sqrt{29-\sqrt{720}}=\sqrt{(2\sqrt{5})^2+3^2-2\sqrt{180}}=2\sqrt 5-3\\ \sqrt{27+\sqrt{648}}=\sqrt{(3\sqrt{2})^2+3^2+2\sqrt{162}}=3\sqrt 2+3\\\sqrt{25-\sqrt{504}}=\sqrt{(3\sqrt{2})^2+(\sqrt 7)^2-2\sqrt{126}}=3\sqrt 2-\sqrt 7\\\sqrt{23-\sqrt{448}}=\sqrt{(\sqrt{7})^2+4^2-2\sqrt{112}}=4-\sqrt 7\\\sqrt{21-\sqrt{80}}=\sqrt{(2\sqrt{5})^2+1^2-2\sqrt{20}}=2\sqrt 5-1}$

Substituting these values in expression we get : $\mathcal{R}=\sqrt{11+(2\sqrt 5-3)+(3\sqrt 2+3)-( 3\sqrt 2-\sqrt 7)+(4-\sqrt 7 )-(2\sqrt 5-1 )}$ $\large =\sqrt{11+5}=\boxed{\color{#0C6AC7}{4}}$

Now, the trick to this question is to rewrite $\sqrt{a \pm \sqrt{b}}=\sqrt{c} \pm \sqrt{d}$ (Or, you can just grab your calculator, but that's no fun)

$\sqrt{29-\sqrt{720}}=\sqrt{p}-\sqrt{q}\\ 29-\sqrt{720}=p+q-2\sqrt{pq}$

Compare the rational and irrational parts, and you get

$29=p+q\implies$ Eq.(1)

$-\sqrt{720}=-2\sqrt{pq}\\ 720=4pq\\ pq=180\\ q=\dfrac{180}{p}$

Substitute this into Eq.(1):

$29=p+\dfrac{180}{p}\\ p^2-29p+180=0\\ (p-20)(p-9)=0\\ p=20,\;p=9$

If $p=9,\;q=20,\;\sqrt{p}-\sqrt{q}<0$ , which is impossible. Therefore, $p=20,\;q=9$ .

$\implies \sqrt{29-\sqrt{720}}=\sqrt{20}-\sqrt{9}$

Now I'm feeling lazy today, so you do the rest yourselves. You should get

$\sqrt{27+\sqrt{648}}=\sqrt{9}+\sqrt{18}\\ \sqrt{25-\sqrt{504}}=\sqrt{18}-\sqrt{7}\\ \sqrt{23-\sqrt{448}}=\sqrt{16}-\sqrt{7}\\ \sqrt{21-\sqrt{80}}=\sqrt{20}-\sqrt{1}$

Substitute all these into the expression:

$\mathcal R = \sqrt{\sqrt{121}+\sqrt{29-\sqrt{720}}+\sqrt{27+\sqrt{648}}- \sqrt{25-\sqrt{504}}+ \sqrt{23-\sqrt{448}}- \sqrt{21-\sqrt{80}}}\\ =\sqrt{\sqrt{121}+(\sqrt{20}-\sqrt{9})+(\sqrt{9}+\sqrt{18})-(\sqrt{18}-\sqrt{7})+(\sqrt{16}-\sqrt{7})-(\sqrt{20}-\sqrt{1})}\\ =\sqrt{\color{magenta}{11}\color{#3D99F6}{+\sqrt{20}}\color{#D61F06}{-3+3}\color{#EC7300}{+\sqrt{18}-\sqrt{18}}\color{#20A900}{+\sqrt{7}}\color{magenta}{+4}\color{#20A900}{-\sqrt{7}}\color{#3D99F6}{-\sqrt{20}}+\color{magenta}{1}}\\ =\sqrt{\color{magenta}{16}}\\ =\boxed{4}$