CAN googlogisms

Here are some googologisms for cursed array notation (here), as well as an analysis.
Let 1 stand for [0], 2 stand for [[0]], etc., and ω stand for [0,[0]].
CAN Level of array Number Name
1$0 0 10 Dek
2$0 100 Duodek
3$0 1 000 Tridek
1$1 1 10^10 Dekti
2$1 10^100 Duodekti
1$2 2 10^10^10 Dektiduo
2$2 10^10^100 Duo-dektiduo
1$3 3 10^10^10^10 Dekatri
1$4 4 10^^5 Dekatetri
1$5 5 10^^6 Dekapenti
1$ω ω 10^^10 Deka-finized
2$ω 10^^100 Duo-dekafinized
1$[1,1] ω+1 10^^10^10 Deka-unfinized
2$[1,1] 10^^10^100 Duo-deka-unfinized
1$[2,1] ω+2 10^^10^10^10 Deka-duofinized
1$[3,1] ω+3 10^^10^10^10^10 Deka-trifinized
1$[ω,1] ω2 10^^10^^10 Deka-finifinized
2$[ω,1] 10^^10^^10^10 Duo-deka-finifinized
1$[[1,1],1] ω2+1 10^^10^^10^10 Deka-unfinifinized
1$[[ω,1],1] ω3 10^^10^^10 Deka-finifinifinized
1$[0,2] ω2 10^^^10 Deka-finidized
2$[0,2] 10^^^100 Duo-deka-finidized
1$[1,2] ω2+1 10^^^10^10 Deka-unfinidized
1$[2,2] ω2+2 10^^^10^10^10 Deka-duofinidized
1$[ω,2] ω2 10^^^10^^10 Deka-finizedfinidized
1$[[1,1],2] ω2+ω+1 10^^^10^^10^10 Deka-unfinizedfinidized
1$[[2,1],2] ω2+ω+2 10^^^10^^10^10^10 Deka-duofinizedfinidized
1$[[ω,1],2] ω2+ω2 10^^^10^^10^^10 Deka-finifinizedfinidized
1$[[0,2],2] ω22 10^^^10^^^10 Deka-finifinidized
1$[[1,2],2] ω22+1 10^^^10^^^10^10 Deka-unfinifinidized
1$[[ω,2],2] ω22+ω 10^^^10^^^10^^10 Deka-finizedfinifinidized
1$[[[0,2],2],2] ω23 10^^^10^^^10^^^10 Deka-finifinifinidized
1$[0,3] ω3 10^^^^10 Deka-finitrized
1$[1,3] ω3+1 10^^^^10^10 Deka-unfinitrized
1$[ω,3] ω3 10^^^^10^^10 Deka-finizedfinitrized
1$[[ω,1],3] ω3+ω2 10^^^^10^^10^^10 Deka-finifinizedfinitrized
1$[[0,2],3] ω32 10^^^^10^^^10 Deka-finidizedfinitrized
1$[[[0,2],2],3] ω322 10^^^^10^^^10^^^10 Deka-finifinidizedfinitrized
1$[[0,3],3] ω32 10^^^^10^^^^10 Deka-finifinitrized
1$[0,4] ω4 10{5}10 Deka-finitetrized
1$[[0,3],4] ω43 10{5}10^^^^10 Deka-finitrizedfinitetrized
1$[[0,4],4] ω42 10{5}10{5}10 Deka-finifinitetrized
1$[0,5] ω5 10{6}10 Deka-finipentized
1$[0,0,1] ωω 10{10}10 Deka-finilized
2$[0,0,1] 10{100}10 Duo-deka-finilized
1$[1,0,1] ωω+1 10{1010}10 Deka-unfinilized
1$[2,0,1] ωω+2 10{101010}10 Deka-duofinilized
1$[ω,0,1] ωω 10{10^^10}10 Deka-finizedfinilized
1$[[1,1],0,1] ωω+ω+1 10{10^^1010}10 Deka-unfinizedfinilized
1$[[ω,1],0,1] ωω+ω2 10{10^^10^^10}10 Deka-finifinizedfinilized
1$[[0,2],0,1] ωω2 10{10^^^10}10 Deka-finidizedfinilized
1$[[0,3],0,1] ωω3 10{10^^^^10}10 Deka-finifinitrizedfinilized
1$[[0,0,1],0,1] ωω2 10{10{10}10}10 Deka-finifinilized
1$[[1,0,1],0,1] ωω2+1 10{10{1010}10}10 Deka-unfinifinilized
1$[[ω,0,1],0,1] ωω2+ω 10{10{10^^10}10}10 Deka-finizedfinifinilized
1$[[[0,0,1],0,1],0,1] ωω3 10{10{10{10}10}10}10 Deka-finilizedfinifinilized
1$[0,1,1] ωω+1 10{{1}}10 Deka-finiduolized
1$[1,1,1] ωω+1+1 10{{1}}1010 Deka-unfiniduolized
1$[ω,1,1] ωω+1 10{{1}}1010 Deka-finizedfiniduolized
1$[[0,0,1],1,1] ωω+1ω 10{{1}}10{10}10 Deka-finilizedfiniduolized
1$[[0,1,1],1,1] ωω+12 10{{1}}10{{1}}10 Deka-finiduolizedfiniduolized
1$[0,2,1] ωω+2 10{{2}}10 Deka-finitrilized
1$[0,3,1] ωω+3 10{{3}}10 Deka-finitetrilized
1$[0,ω,1] ωω2 10{{10}}10 Deka-finifenlized
1$[[0,ω,1],ω,1] ωω22 10{{10{{10}}10}}10 Deka-finifenlizfinifenlized
1$[0,[1,1],1] ωω2+1 10{{{1}}}10 Deka-finiduofenlized
1$[0,[2,1],1] ωω2+2 10{{{2}}}10 Deka-finitrifenlized
1$[0,[ω,1],1] ωω3 10{{{10}}}10 Deka-finifefenlized
1$[0,[[1,1],1],1] ωω3+1 10{{{{1}}}}10 Deka-finidefefenlized
1$[0,[[ω,1],1],1] ωω4 10{{{{10}}}}10 Deka-finifefefenlized
1$[0,0,2] ωω2 {10,10,1,10} Deka-finiledetized
1$[1,0,2] ωω2+1 {10,10,1,1010} Deka-unfiniledetized
1$[[0,1],0,2] ωω2 {10,10,1,10^^10} Deka-finizedfiniledetized
1$[[0,2],0,2] ωω22 {10,10,1,10^^^10} Deka-finidizedfiniledetized
1$[[0,0,1],0,2] ωω2ω {10,10,1,{10,10,10}} Deka-finilizedfiniledetized
1$[[0,1,1],0,2] ωω2ω+1 {10,10,1,{10,10,1,2}} Deka-finidelizedfiniledetized
1$[[0,ω,1],0,2] ωω2ω2 {10,10,1,{10,10,10,2}} Deka-finifenlizedfiniledetized
1$[[0,0,2],0,2] ωω22 {10,10,1,{10,10,1,10}} Deka-finifiniledetized
1$[0,1,2] ωω2+1 {10,10,1,1,2} Deka-finiduoledetized
1$[0,2,2] ωω2+2 {10,10,2,1,2} Deka-finiduoledetized
1$[0,ω,2] ωω2 {10,10,10,1,2} Deka-finifenledetized
1$[0,[1,1],2] ωω2+ω+1 {10,10,1,2,2} Deka-finiduofenledetized
1$[0,[ω,1],2] ωω2+ω2 {10,10,10,2,2} Deka-finifefenledetized
1$[0,[0,2],2] ωω22 {10,10,1,10,2} Deka-finifidienledetized
1$[0,[1,2],2] ωω22+1 {10,10,1,1,3} Deka-finiduofidienledetized
1$[0,[ω,2],2] ωω22+ω {10,10,10,1,3} Deka-finifefidienledetized
1$[0,[[1,1],2],2] ωω22+ω+1 {10,10,1,2,3} Deka-finiduofefidienledetized
1$[0,[[ω,1],2],2] ωω22+ω2 {10,10,10,2,3} Deka-finifefefidienledetized
1$[0,[[0,2],2],2] ωω23 {10,10,1,10,3} Deka-finifefidienledetized
1$[0,0,3] ωω3 {10,10,1,1,10} Deka-finiletretized
1$[0,1,3] ωω3+1 {10,10,1,1,1,2} Deka-finiduoletretized
1$[0,2,3] ωω3+2 {10,10,2,1,1,2} Deka-finitriletretized
1$[0,ω,3] ωω3 {10,10,10,1,1,2} Deka-finifenletretized
1$[0,[1,1],3] ωω3+ω+1 {10,10,1,2,1,2} Deka-finiduofenletretized
1$[0,[ω,1],3] ωω3+ω2 {10,10,10,2,1,2} Deka-finifefenletretized
1$[0,[0,2],3] ωω32 {10,10,1,10,1,2} Deka-finifidienletretized
1$[0,[1,2],3] ωω32+1 {10,10,1,1,2,2} Deka-finiduofidienletretized
1$[0,[ω,2],3] ωω32 {10,10,10,1,2,2} Deka-finifenfidienletretized
1$[0,[[ω,1],2],3] ωω32+ω2 {10,10,10,2,2,2} Deka-finifefenfidienletretized
1$[0,[[0,2],2],3] ωω322 {10,10,1,10,2,2} Deka-finifefidienletretized
1$[0,[0,3],3] ωω32 {10,10,1,1,10,2} Deka-finifitrienletretized
1$[0,[[0,3],3],3] ωω33 {10,10,1,1,10,3} Deka-finifefitrienletretized
1$[0,0,4] ωω4 {10,10,1,1,1,10} Deka-finiletetretized
1$[0,1,4] ωω4+1 {10,10,1,1,1,1,2} Deka-finiduoletetretized
1$[0,[0,4],4] ωω42 {10,10,1,1,1,10,2} Deka-finifitetrienletetretized
1$[0,0,5] ωω5 {10,10,1,1,1,1,10} Deka-finilepentetized
1$[0,0,0,1] ωωω {10,10[2]2} Deka-finikized