2 added 20 characters in body
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Mathematica uses a ComplexityFunction to measure how 'simple' an expression is. According to the documentation:

ComplexityFunction counts the subexpressions and digits of integers

What we need to do is create a ComplexityFunction that considers the expressions you desire to be very cheap. Depth almost works for your middle expression

f[x_] := Depth[x];
FullSimplify[{1/(1 + Sqrt[2] - Sqrt[3]), Sqrt[5 - 2*Sqrt[6]], 
  Sqrt[4 + Sqrt[15]]}, ComplexityFunction -> f]

gives

enter image description here

Sadly, the expression you want has the same depth as the expression you started with

Depth[Sqrt[4 + Sqrt[15]]]

4

 Depth[1/2 (Sqrt[6] + Sqrt[10])]

 4

I'm afraid that I couldn't find anything that would Simplify that final expression.

Mathematica uses a ComplexityFunction to measure how 'simple' an expression is. According to the documentation:

ComplexityFunction counts the subexpressions and digits of integers

What we need to do is create a ComplexityFunction that considers the expressions you desire to be very cheap. Depth almost works

f[x_] := Depth[x];
FullSimplify[{1/(1 + Sqrt[2] - Sqrt[3]), Sqrt[5 - 2*Sqrt[6]], 
  Sqrt[4 + Sqrt[15]]}, ComplexityFunction -> f]

gives

enter image description here

Sadly, the expression you want has the same depth as the expression you started with

Depth[Sqrt[4 + Sqrt[15]]]

4

 Depth[1/2 (Sqrt[6] + Sqrt[10])]

 4

I'm afraid that I couldn't find anything that would Simplify that final expression.

Mathematica uses a ComplexityFunction to measure how 'simple' an expression is. According to the documentation:

ComplexityFunction counts the subexpressions and digits of integers

What we need to do is create a ComplexityFunction that considers the expressions you desire to be very cheap. Depth works for your middle expression

f[x_] := Depth[x];
FullSimplify[{1/(1 + Sqrt[2] - Sqrt[3]), Sqrt[5 - 2*Sqrt[6]], 
  Sqrt[4 + Sqrt[15]]}, ComplexityFunction -> f]

gives

enter image description here

Sadly, the expression you want has the same depth as the expression you started with

Depth[Sqrt[4 + Sqrt[15]]]

4

 Depth[1/2 (Sqrt[6] + Sqrt[10])]

 4

I'm afraid that I couldn't find anything that would Simplify that final expression.

1
source | link

Mathematica uses a ComplexityFunction to measure how 'simple' an expression is. According to the documentation:

ComplexityFunction counts the subexpressions and digits of integers

What we need to do is create a ComplexityFunction that considers the expressions you desire to be very cheap. Depth almost works

f[x_] := Depth[x];
FullSimplify[{1/(1 + Sqrt[2] - Sqrt[3]), Sqrt[5 - 2*Sqrt[6]], 
  Sqrt[4 + Sqrt[15]]}, ComplexityFunction -> f]

gives

enter image description here

Sadly, the expression you want has the same depth as the expression you started with

Depth[Sqrt[4 + Sqrt[15]]]

4

 Depth[1/2 (Sqrt[6] + Sqrt[10])]

 4

I'm afraid that I couldn't find anything that would Simplify that final expression.