Ruby, 55 53 47 45 3637 bytes
f=->s,a=1{s&&[s[0,a],f[s[a*f[s[a..],a+a]]}
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@OP just to let you know, the easy tasks are welcome! :)
A recursive function with 2 variables: the original string and the counter. The function chops a string piece, adds it to the unnamed output vector and updates the counter until the string is empty.