There are such versions of the law of the iterated logarithm even for independent random vectors in an arbitrary separable Banach space. See e.g. [Theorems 4.1 and 4.2][1]. 

In the case when the Banach space is a Hilbert one, [Theorem 4.1][1] implies the following: 

>Let $X,X_1,X_2,\dots$ be iid random vectors in a separable Hilbert space $(H,\langle\cdot,\cdot\rangle,|\cdot|)$ with $EX=0$ and $E|X|^2<\infty$. Let $S_n:=X_1+\cdots+X_n$. Then 
$$\limsup_n\frac{|S_n|}{\sqrt{2n\ln\ln n}}=\sigma$$
almost surely, where 
$$\sigma:=\sup\big\{\sqrt{E\langle X,f\rangle^2}\colon f\in H,|f|=1|\big\}.$$

To obtain this from [Theorem 4.1][1], one only needs to note the following two points: 

(i) In view of [formula (2.5)][2], $\sup_{x\in K}|x|=\sigma$.

(ii) $E|S_n|\le\sqrt{E|S_n|^2}=\sqrt{nE|X|^2}=o(\sqrt{2n\ln\ln n})$, so that condition (ii) of [Theorem 4.1][1] holds. 


  [1]: https://projecteuclid.org/euclid.ijm/1256048928
  [2]: https://projecteuclid.org/euclid.aop/1176995982