The nonlinear single and double ionization of Sr atoms as a result of their interaction with 800 nm Ti-Sapphire laser pulses of 25 fs duration is experimentally investigated within the laser intensity range I = 3 TWcm -2 -30 TWcm -2 . The latter range includes the over-the-barrier intensities for both Sr and Sr + , while the corresponding Keldysh adiabaticity parameters are larger than unity.Nevertheless, for I≥24 TWcm -2 the recorded photoelectron energy spectra show evidence of the (partial) applicability of tunneling or over-the-barrier ionization concepts, whereas below this value they can be discussed in terms of multiphoton ionization processes. More importantly, the data additionally point towards a further division of the multiphoton regime into two sub-ranges separated roughly at I ~ 4 TWcm -2 . Above this value above-threshold-ionization structures and the single-active-electron picture dominate. On the contrary, at the lowest laser intensities the photoelectron spectra and angular distributions reveal single ionization pathways where the remaining ion is left into excited states. These pathways also require photon absorption in the continuum, which in the case of strontium and other alkaline earth atoms is structured by the presence of doubly excited states which are embedded in it. Hence, in this case ionization proceeds via quasi-resonant ionization ladders formed by these doubly excited states which are dominated by the interaction between the two valence electrons of Sr. We discuss in detail the above rich phenomenology and relevant ionization mechanisms and propose possible directions of further work towards the elucidation of the role of configuration interaction in ionization processes.