Transform sphere into a cube

Slow, but it works:

Animate[
 RegionPlot3D[
  With[{u = Sin[t]^2*10 + 2}, 
   Abs[x]^u + Abs[y]^u + Abs[z]^u < 1], {x, -1, 1}, {y, -1, 
   1}, {z, -1, 1}, PerformanceGoal -> "Quality"], {t, 0, \[Pi]}]

reg = DiscretizeRegion[Cuboid[{-1, -1, -1}, {1, 1, 1}], 
   MaxCellMeasure -> .01];
DynamicModule[{pts = MeshCoordinates[reg], 
  norms = Norm /@ MeshCoordinates[reg]}
 , Animate[
  Graphics3D@GraphicsComplex[
    Dynamic[ pts  /(1 - t + t  norms) ],
    {EdgeForm@None, MeshCells[reg, {2}]}
    ]
  , {t, 0, 1}, AnimationRate -> 1, 
  AnimationDirection -> ForwardBackward]
 ]

One possibility is to transform : 1) the Sphere to a cow 2) then the cow to a cube

cow = ExampleData[{"Geometry3D", "Cow"}];
Join[
Table[cow /. GraphicsComplex[array1_, rest___] :>  
                  GraphicsComplex[(# (Norm[#]^-coeff)) & /@ array1,rest],{coeff,1,0,-.2}],
Table[cow /. GraphicsComplex[array1_, rest___] :>  
                  GraphicsComplex[Map[(# (Norm[#]^-coeff)) & ,array1,{2}], rest],{coeff,0,1,.2}]
] //Multicolumn[#,Appearance-> "Horizontal"]&

inspiration source