  Let $p\ge2$, $n_1\le...\le n_p$ be positive integers and $B_1^1, ...,
B_{n_1}^1; ...; B_1^p, ..., B_{n_p}^{p}$ be independent planar Brownian motions
started uniformly on the boundary of the unit circle. We define a $p$-fold
intersection exponent $\varsigma(n_1,..., n_p)$, as the exponential rate of
decay of the probability that the packets $\bigcup_{j=1}^{n_i} B_j^i[0,t^2]$,
$i=1,...,p$, have no joint intersection. The case $p=2$ is well-known and,
following two decades of numerical and mathematical activity, Lawler, Schramm
and Werner (2001) rigorously identified precise values for these exponents. The
exponents have not been investigated so far for $p>2$. We present an extensive
mathematical and numerical study, leading to an exact formula in the case
$n_1=1$, $n_2=2$, and several interesting conjectures for other cases.
