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In 2011, Dekel et al. developed highly geometric Hardy spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>p</mi></msup><mrow><mo>(</mo><mo>Θ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, for the full range <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>p</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, which were constructed by continuous multi-level ellipsoid covers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Θ</mo></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula> with high anisotropy in the sense that the ellipsoids can rapidly change shape from point to point and from level to level. In this article, when the ellipsoids in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Θ</mo></semantics></math></inline-formula> rapidly change shape from level to level, the authors further obtain some real-variable characterizations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>p</mi></msup><mrow><mo>(</mo><mo>Θ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in terms of the radial, the non-tangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaces of Bownik.