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 Halász and proving a conjecture of Erdős. 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.