formatting: extract outputs from a WHILE cycle in table format

Use Table instead of a While tie:

table = Table[{2, 3}^i, {i, 5}]

{{2, 3}, {4, 9}, {8, 27}, {16, 81}, {32, 243}}

OR

table = Transpose[Table[i^Range[5], {i, 2, 3}]]

{{2, 3}, {4, 9}, {8, 27}, {16, 81}, {32, 243}}

You can use Grid to show:

Grid[table, Dividers -> All] 

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To update: If you have to use While you can AppendTo a list that is initialized in {}:

i = 1;
tab = {};
While[i < 6, AppendTo[tab, {2, 3}^i++]]
tab

{{2, 3}, {4, 9}, {8, 27}, {16, 81}, {32, 243}}

Alternatively, you can use Reap / / Sow combination:

i = 1;
Reap[While[i < 6, Sow[ {2, 3}^i++]]][[2, 1]]

{{2, 3}, {4, 9}, {8, 27}, {16, 81}, {32, 243}}