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The turbulent small-scale dynamo (SSD) is likely to be responsible for the magnetisation of the interstellar medium (ISM) that we observe in the Universe today. The SSD efficiently converts kinetic energy $E_{\rm kin}$ into magnetic energy $E_{\rm mag} $, and is often used to explain how an initially weak magnetic field with $E_{\rm mag} \ll E_{\rm kin}$ is amplified, and then maintained at a level $E_{\rm mag} \lesssim E_{\rm kin}$. Usually, this process is studied by initialising a weak seed magnetic field and letting the turbulence grow it to saturation. However, in this Part I of the Growth or Decay series, using three-dimensional, visco-resistive magnetohydrodynamical turbulence simulations up to magnetic Reynolds numbers of 2000, we show that the same final state in the integral quantities, energy spectra, and characteristic scales of the magnetic field can also be achieved if initially $E_{\rm mag} \sim E_{\rm kin}$ or even if initially $E_{\rm mag} \gg E_{\rm kin}$. This suggests that the final saturated state of the turbulent dynamo is set by the turbulence and the material properties of the plasma, independent of the initial structure or amplitude of the magnetic field. We discuss the implications this has for the maintenance of magnetic fields in turbulent plasmas and future studies exploring the dynamo saturation., Comment: 16 pages. 11 figures. Accepted to MNRAS. Part I focuses on the journey towards saturation. Part II focuses on the decay into saturation