I create the notation extension based off Taranovsky's notation and the fast-growing hierarchy, that grows much faster than the Tar(n) function itself as a naive extension. There are at least three levels of extension.
Basic level [Format: XTar(a; n)]
Recurring level [Format: Xtar(b, a; n)]
Finitary level [Format: Xtar(..., d, c, b, a; n)]
Transfinitary level [Format: Xtar(... {k3} c {k2} b {k1} a; n)]
Collapsing level [Format: TBD]
It has two-variable format: XTar(a; n). This level can easily beat Denis' "tarintar"!
This level is defined similarly to that of the fast-growing hierarchy, but it's much stronger, and can beat Denis' "tarintar".
Base rule: XTar(n) = XTar(1; n) = Tar(n) = f_{C(C(C(... C(C(C(Ω_n · 2), 0), 0) ...), 0), 0)}(n) with a copies of C, using the fast-growing hierarchy with fundamental sequences for Taranovsky's notation
Degeneration rule: XTar(0; n) = n
Iteration rule: XTar(a+1; n) = XTar^n(a; n), where XTar^a denotes function iteration, and a >= 1
Limit rule: XTar(α; n) = XTar(α[n], n) if α is a limit ordinal
Here, α[n] denotes the n-th term of a fixed fundamental sequence assigned to ordinal α. A system of fundamental sequences for limit ordinals below a given supremum is not unique, and the extensible Tar hierarchy heavily depends on the choice of such a system, as we explained in the introduction to the fast-growing hierarchy. The fundamental sequence associated with this hierarchy is the same as in the fast-growing hierarchy.
XTar(1; 3) = XTar(3) = Tar(3) = f_{C(C(C(Ω_3, 0), 0), 0)}(3) "tritar"
XTar(2; 10) = XTar(10) = Tar(10) "dekotar"
XTar(0; 100) = 100
XTar^2(1; 10) = XTar(1; XTar(10)) = Tar(Tar(10)) "unintar"
XTar(2; 10) = XTar^10(1; 10) = XTar(1; XTar(1; XTar(1; ... XTar(1; XTar(1; 10)) ...))) w/ 10 XTar's = Tar(Tar(Tar(Tar(Tar(Tar(Tar(Tar(Tar(Tar(10))))))))) (10 Tar's) "nonintar"
XTar(2; XTar(1; 10)) = XTar^{XTar(1; 10)}(1; XTar(1; 10)) = XTar^{XTar(1; 10) + 1}(1; 10) = XTar(1; XTar(1; XTar(1; ... XTar(1; XTar(1; 10)) ...))) w/ XTar(1; 10) + 1 XTar's = Tar(Tar(Tar(... Tar(Tar(Tar(10))) ...))) w/ Tar(10) + 1 Tar's "tarintar"
XTar(3; 10) = XTar^10(2; 10) = XTar(2; XTar(2; XTar(2; ... XTar(2; XTar(1; 10)) ...))) w/ 10 XTar's
XTar(ω; 10) = XTar(ω[10]; 10) = XTar(10; 10) = XTar^10(9; 10)
XTar(ω+1; 10) = XTar^10(ω; 10) = XTar(ω; XTar(ω; XTar(ω; ... XTar(ω; XTar(10; 10)) ...))) w/ 10 XTar's
XTar(ω2; 10) = XTar(ω2[10]; 10) = XTar(ω+10; 10) = XTar^10(ω+9; 10)
XTar(ω^2; 10) = XTar(ω^2[10]; 10) = XTar(ω10; 10) = XTar(ω9+10; 10)
XTar(ω^3; 10) = XTar(ω^3[10]; 10) = XTar(ω^2*10; 10)
XTar(ω^ω; 10) = XTar(ω^ω[10]; 10) = XTar(ω^10; 10)
XTar(ω^ω^ω; 10) = XTar(ω^ω^ω[10]; 10) = XTar(ω^ω^10; 10)
XTar(ε0; 10) = XTar(ε0[10]; 10) = XTar(ω^ω^ω^ω^ω^ω^ω^ω^ω; 10)
XTar(ζ0; 10) = XTar(ζ0[10]; 10)
XTar(Γ0; 10) = XTar(Γ0[10]; 10)
XTar(LVO; 10) = XTar(ψ0(Ω^Ω^Ω); 10)
XTar(BHO; 10) = XTar(ψ0(Ω_2); 10)
XTar(BO; 10) = XTar(ψ0(Ω_ω); 10) = XTar(ψ0(Ω_10); 10)
XTar(TFBO; 10) = XTar(ψ0(Ω_(ω+1)); 10)
XTar(EBO; 10) = XTar(Φ1(0); 10)
This level introduces the second entry. It has three-variable format: XTar(b, a; n)
Base rule: XTar(n) = XTar(1; n) = XTar(0, 1; n) = Tar(n) = f_{C(C(C(... C(C(C(Ω_n · 2), 0), 0) ...), 0), 0)}(n) with a copies of C, using the fast-growing hierarchy with fundamental sequences for Taranovsky's notation
Degeneration rule: XTar(0; n) = XTar(b, 0; n) = n
Iteration rule: XTar(b, a+1; n) = XTar^n(b, a; n), where XTar^a denotes function iteration, and a >= 1
Carrying rule: XTar(b+1, a; n) = XTar(b, C(C(C(... C(C(C(Ω_n · 2), 0), 0) ...), 0), 0); n) with a copies of C, using the fast-growing hierarchy with fundamental sequences for Taranovsky's notation
Limit rule: XTar(b, α; n) = XTar(b, α[n]; n), XTar(α, 1; n) = XTar(α[n], 1; n); if α is a limit ordinal
Here α[n] is explained above.
XTar(0, 1; 10) = XTar(1; 10) = Tar(10) "dekotar"
XTar(2, 0; 10) = XTar(0; 10) = 10
XTar(1, 3; 3) = XTar^3(1, 2; 3) = XTar(1, 2; XTar(1, 2; XTar(1, 2; 3)))
XTar(1, 1; 10) = XTar(0, C(C(C(... C(C(C(Ω_10 · 2), 0), 0) ...), 0), 0); 10) with 10 C's = XTar(C(C(C(... C(C(C(Ω_10 · 2), 0), 0) ...), 0), 0); 10) with 10 C's
XTar(ω, 1; 10) = XTar(10, 1; 10) = XTar(9, C(C(C(... C(C(C(Ω_10 · 2), 0), 0) ...), 0), 0); 10) with 10 C's
XTar(ε0, 1; 10) = XTar(ε0[10], 1; 10) = XTar(ω^ω^ω^ω^ω^ω^ω^ω^ω, 1; 10)