In　this　paper,　we　study　a　Leslie-Gower　predator-prey　model　with　strong　Allee　effects　and　constant　prey　refuges.　It　is　shown　that　the　model　can　undergo　a　cusp　type　degenerate　Bogdanov-Takens　bifurcation　of　codimension　4,　focus　and　elliptic　types　degenerate　Bogdanov-Takens　bifurcations　of　codimension　3,　and　degenerate　Hopf　bifurcation　of　codimension　3　as　the　parameters　vary.　The　model　can　exhibit　the　coexistence　of　multiple　positive　steady　states,　multiple　limit　cycles,　and　homoclinic　loops.　Our　results　indicate　that　a　larger　prey　refuge　contributes　to　the　coexistence　of　both　species.　Numerical　simulations,　including　three　limit　cycles,　quadristability,　a　large-amplitude　limit　cycle　enclosing　three　positive　steady　states　and　a　homoclinic　loop,　two　large-amplitude　limit　cycles　enclosing　three　positive　steady　states,　are　presented　to　illustrate　the　theoretical　results.