Let $X_1, X_2, \ldots $ be a sequence of independent identically distributed random variables with values in the space $C([0,1])$ of real continuous functions on $[0,1$]. Assume for simplicity that the $X_i$'s are centered, in the sense that $E( X_i(t)) = 0$ for $t \in [0,1]$). Are there any necessary and sufficient conditions (on the common distribution of the $X_i$'s) in order that $$ \limsup_{n \to \infty} \frac {1}{\sqrt {2n \ln \ln (n)}} \, \| X_1 + \cdots +X_n \|_\infty < \infty $$ almost surely (where $\| \cdot \|_\infty$ is the sup norm)? What would then be the value of the limsup?

What you want is the so-called bounded law of the iterated logarithm (BLIL) in $C[0,1]$.

A necessary and sufficient condition for the BLIL in an arbitrary real separable Banach space $B$ was given by Ledoux and Talagrand '88 -- see e.g. Theorem A in the subsequent paper by Einmahl '93. This necessary and sufficient condition involves the boundedness of $(S_n/\sqrt{n\ln\ln n})_{n\ge3}$ in probability, which may be hard to verify in $C[0,1]$.

A sufficient condition (involving a certain, rather weak condition on the modulus of continuity of $X_1(\cdot)$) for the BLIL in $C[0,1]$ is implied by Theorem 2.1 in the earlier paper by Kuelbs '76.