There are a lot of algorithms that you can do on a Rubik's cube.  These are a few of my favorites.

Cube Patterns

​Simple checkered pattern

M2 E2 S2 (or move every other side on larger cubes)

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Flower Pattern

E M E' M' (move multiple internal pieces for larger cubes)

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Advanced checkered pattern

F B2 R' D2 (B R U D' R L' D' F') R2 D F2 B'

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Cube in a cube
 - Outside algorithm - (R2 F2 U2)  R D2 L'  (B' U)3x  L D2 R

 - Inside algorithm - (R' D' R D)2x U (R' D' R D)4x   x'   (R' D' R D)2x U2 (R' D' R D)4x​​

          or easier version -L2  (R' D' R D)2x U' (D' R' D R)2x U  L2

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Superflip (Flip all edges)

(M U')x4  y z'  (M U')x4  y z'  (M U')x4
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For larger cubes start with the outside layers and move in (So for the 7x7 pictured do the M move by moving all 5 internal layers down, then U' on the top layer.  Then move in by doing the M move with the inner 3 layers and U' with the top 2 layers.