  We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative
function, there almost surely exist arbitrarily large values of $x$ for which
$|\sum_{n \leq x} f(n)| \geq \sqrt{x} (\log\log x)^{1/4+o(1)}$. This is the
first such bound that grows faster than $\sqrt{x}$, answering a question of
Hal\'asz and proving a conjecture of Erd\H{o}s. It is plausible that the
exponent $1/4$ is sharp in this problem.
  The proofs work by establishing a multivariate Gaussian approximation for the
sums $\sum_{n \leq x} f(n)$ at a sequence of $x$, conditional on the behaviour
of $f(p)$ for all except the largest primes $p$. The most difficult aspect is
showing that the conditional covariances of the sums are usually small, so the
corresponding Gaussians are usually roughly independent. These covariances are
related to an Euler product (or multiplicative chaos) type integral twisted by
additive characters, which we study using various tools including mean value
estimates for Dirichlet polynomials, high mixed moment estimates for random
Euler products, and barrier arguments with random walks.
