We　propose　a　parameter-free　and　locking-free　enriched　Galerkin　method　of　arbitrary　order　for　solving　the　linear　elasticity　problem　in　both　two　and　three　space　dimensions.　Our　method　uses　an　approximation　space　that　enriches　the　vector-valued　continuous　Galerkin　space　of　order　k　with　some　discontinuous　piecewise　polynomials.　To　the　best　of　our　knowledge,　it　extends　the　locking-free　enriched　Galerkin　space　in　Yi　et　al.　(2022)　to　high　orders　for　the　first　time.　Compared　to　the　continuous　Galerkin　method,　the　proposed　method　is　locking-free　with　only　k　d(-　1)　additional　degree　of　freedom　on　each　element.　The　parameter-free　property　of　our　method　is　realized　by　integrating　the　enriched　Galerkin　space　into　the　framework　of　the　modified　weak　Galerkin　method.　We　rigorously　establish　the　well-posedness　of　the　method　and　provide　optimal　error　estimates　for　the　compressible　case.　Extensive　numerical　examples　confirm　both　the　accuracy　and　the　locking-free　property　of　the　proposed　method.