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I'd like to create a sphere using a cube as a starting point. The way I'd normally go about that is to add a couple of subsurf modifiers and then apply them, however, this does not give a perfect sphere.

I recall seeing somewhere that somebody used a command after this process to make the rounded cube perfectly spherical, unfortunately I do not remember what this is.

Could somebody please inform me how to create this spherical shape in Blender using this basic technique?

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As you noted correctly, the catmull clark subdivision will not produce spherical shapes.

Use the to sphere command.

The To Sphere transformation will give the selection spherical qualities. The Fig. Monkey with increasing sphericity. below shows the results of applying the To Sphere transformation to the monkey mesh.
suzanne to sphere

Select all geometry A and press ⎈ Ctrl⇧ Shift⎇ AltS

Or locate the menu: Mesh ‣ Transform ‣ To Sphere

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Non-destructive way would be to use Subdivision Surface modifer along with Cast modifier set to Sphere > Factor > 1.

modifier

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In addition to Leander's and cgslav's answers note that before you use the To Sphere command or the Cast modifier you can fiddle with the values in the Subdivide's options panel at the bottom of the Tools on the left. You only need to click on Subdivide once and then you can adjust the Number of Cuts to subdivide it how many times you need, and if you change Smoothness to 1 it makes it spherical.

And if you want to make it bumpy and distressed, try increasing the Fractal value.

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