Closed form or upper bound for recursively defined sequence

Is there a closed form of the following sequence:

$$u_0 = 2$$

$$u_{n+1} = s_n^2-s_n, \;s_n = \sum_{k=0}^{n} u_k$$

If not, I would like to have an upper bound. By looking at the numbers I guessed that $2^{2^n}$ is one, is this true? Is there a better one?

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$u_{n+1}=s_{n+1}-s_n=s_n^2-s_n$, therefore $s_{n+1}=s_n^2$. Hence $\log(s_{n+1})=2\log(s_n)$. Let $b_n=\log(s_n)$. Solve for $b_n$ and then $s_n$. Then find $u_n$:
$$u_n=s_n-s_{n-1}$$