The problems of required conditions and possible consequences of the super - compression (up to $R_m\sim 3R_E$) of the Earth's magnetosphere by giant CME are investigated by the methods of laboratory and computer simulations. A useful relation between an expected magnetopause location $R_m^*$ and the kinetic plasma energy $E_0$ of spherical plasma cloud (exploded at distance $R_0$) was obtained $R_m^*/R_0\approx 0,75/{\ae}^{1/6}$ and tested by MHD – model of Nikitin & Ponomarenko (1994) with the using their main energetic criterion of the problem ${\ae}=3E_0 R_0^3/\mu^2$ (for magnetic moment $\mu$ of point obstacle in vacuum). This relation could describe rather well an observed compression ($R_m\sim 5-6 R_E$ for CME with energy $10^{32}\, ergs$ and effective value $E_0\sim 10^{33} ergs,$ into $4\pi$) and predicts $R_m^*\leq 3R_E$ in a probable case of Mega Flare with the total energy release $\sim 10^{34}\, ergs$ and possible $E_0\sim 5\cdot 10^{34}\, ergs$ according to Kane et al. (1995) and Tsurutani et al. (2003). Some most important features of the formation such Artificial Magnetosphere (AM) structure and its possible influence onto various geospheres media (or technosphere areas) could be successfully studied in the simulative experiments at KI-1 facility of ILP with Laser Plasmas (LP) of $E_0$ up to $kJ$ and dipole $\mu\sim 10^7\, G\cdot cm^3$ as was shown by Ponomarenko et al. (2001) and Zakharov (2003). But the main problem of such planned AMEX experiment (at ${\ae}\sim 50$ for $R_0=75\, cm$) is the influence of finite value of ion magnetization $\varepsilon_m=R_L/R_m^*$ based on the ion Larmor radius $R_L=mcV_0/ezB_d,$ where $V_0\sim 100\, km/s$ is the expansion velocity of LP and $B_d$ is the initial dipole field at the point $R_m^*.$ Of coarse, $\varepsilon_m\ll 1$ in a real space conditions (excluding cases of Mercury or asteroids, explored by Omidi et al. (2004)) while in the laboratory to fulfill both need constrains ${\ae}\gg 1$ and $\varepsilon_m\ll 1$ we should use a thermonuclear plasma and devices. To overcome this problem we did a 3D/PIC – calculations by hybrid model of Kyushu University, described by Muranaka et al. (2001), to find out a critical value of $\varepsilon_m^*$ $(\approx 0,2-\!0,3),$ which need for MHD – like interaction of exploding plasmas with magnetic dipole.