The following are well-known:
$\limsup_{t\rightarrow \infty} \frac{B(t)}{t} = 0$
$\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt t} = \infty$
$\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt {2t\log\log t}} = 1$
What about any of the following:
$\limsup_{t\rightarrow \infty} \int_{0}^{t} \frac{B(s)}{s} ds $
$\limsup_{t\rightarrow \infty} \int_{0}^{t} \frac{B(s)}{\sqrt s} ds $
$\limsup_{t\rightarrow \infty} \int_{0}^{t} \frac{B(s)}{\sqrt {2s\log\log s}} ds $
I wasn't able to find anything about these online, so I'm not sure if these are unsolved or are unsolveable since the integral clearly complicates the problem, but figured there's no harm in asking!