We present a detailed X-ray timing analysis of the highly variable NLS1 galaxy, IRAS 13224-3809. The source was recently monitored for 1.5 Ms with XMM-Newton which, combined with 500 ks archival data, makes this the best studied NLS1 galaxy in X-rays to date. We apply standard time- and Fourier-domain in order to understand the underlying variability process. The source flux is not distributed lognormally, as would be expected for accreting sources. The first non-linear rms-flux relation for any accreting source in any waveband is found, with $\mathrm{rms}\propto \mathrm{flux}2/3$. The light curves exhibit significant strong non-stationarity, in addition to that caused by the rms-flux relation, and are fractionally more variable at lower source flux. The power spectrum is estimated down to $\sim 10-7$ Hz and consists of multiple peaked components: a low-frequency break at $\sim 10-5$ Hz, with slope down to low frequencies; an additional component breaking at $\sim 10-3$ Hz. Using the high-frequency break we estimate the black hole mass $M\mathrm{BH}=[0.5-2]\times 106M\odot$, and mass accretion rate in Eddington units, $m\dot{}Edd\gtrsim 1$. The non-stationarity is manifest in the PSD with the normalisation of the peaked components increasing with decreasing source flux, as well as the low-frequency peak moving to higher frequencies. We also detect a narrow coherent feature in the soft band PSD at $0.7$ mHz, modelled with a Lorentzian the feature has $Q\sim 8$ and an $\mathrm{rms}\sim 3$ %. We discuss the implication of these results for accretion of matter onto black holes.