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R Function created by Hyp cos

$$\{0\}$$ has growth rate $$\omega$$

$$n\{0\}$$ has growth rate $$\omega + n$$

$$\{0\}\{0\}$$ has growth rate $$\omega 2$$

$$\underbrace{\{0\}...\{0\}}_{n}$$ has growth rate $$\omega n$$

$$\{1\}$$ has growth rate $$\omega^2$$

$$\{n\}$$ has growth rate $$\omega^{n-1}$$

$$\{\{0\}\}$$ has growth rate $$\omega^\omega$$

$$\{n\{0\}\}$$ has growth rate $$\omega^{\omega + n}$$

$$\{\underbrace{\{0\}...\{0\}}_{n}\}$$ has growth rate $$\omega^{\omega n}$$

$$\{\{n\}\}$$ has growth rate $$\omega^{\omega^{n-1}}$$

$$\{\{\{0\}\}\}$$ has growth rate $$\omega^\omega$$

$$\underbrace{\{...\{0\}...\}}_{n}$$ has growth rate $$^n\omega$$

$$\{0,1\}$$ has growth rate $$\varepsilon_0$$

$$n\{0,1\}$$ has growth rate $$\varepsilon_0+n$$

$$\underbrace{\{0\}...\{0\}}_{n}\{0,1\}$$ has growth rate $$\varepsilon_0+\omega n$$

$$\{n\}\{0,1\}$$ has growth rate $$\varepsilon_0+\omega^{n+1}$$

$$\underbrace{\{...\{0\}...\}}_{n}\{0,1\}$$ has growth rate $$\varepsilon_0+^n\omega$$

$$\{\{0,1\}\}\{0,1\}$$ has growth rate $$\varepsilon_0 2$$

$$\{1\{0,1\}\}\{0,1\}$$ has growth rate $$\varepsilon_0 \omega$$

$$\{\{\{0,1\}\}\{0,1\}\}\{0,1\}$$ has growth rate $$\varepsilon_0^2$$

$$\{\{\{\{0,1\}\}\{0,1\}\}\{0,1\}\}\{0,1\}$$ has growth rate $$\varepsilon_0^{\varepsilon_0}$$

$$\{0,1\}\{0,1\}$$ has growth rate $$\varepsilon_1$$

$$\{\{0,1\}\}\{0,1\}\{0,1\}$$ has growth rate $$\varepsilon_1+\varepsilon_0$$

$$\{\{0,1\}\{0,1\}\}\{0,1\}\{0,1\}$$ has growth rate $$\varepsilon_1 2$$

$$\{\{\{0,1\}\{0,1\}\}\{0,1\}\{0,1\}\}\{0,1\}\{0,1\}$$ has growth rate $$\varepsilon_1^2$$

$$\{\{\{\{0,1\}\{0,1\}\}\{0,1\}\{0,1\}\}\{0,1\}\{0,1\}\}\{0,1\}\{0,1\}$$ has growth rate $$\varepsilon_1^{\varepsilon_1}$$

$$\{0,1\}\{0,1\}\{0,1\}$$ has growth rate $$\varepsilon_2$$

$$\underbrace{\{0,1\}...\{0,1\}}_{n}$$ has growth rate $$\varepsilon_{n-1}$$

$$\{1,1\}$$ has growth rate $$\varepsilon_\omega$$

$$\{n,1\}$$ has growth rate $$\varepsilon_{\omega^n}$$

$$\{\{0\},1\}$$ has growth rate $$\varepsilon_{\omega^\omega}$$

$$\{\{\{0,1\}\},1\}$$ has growth rate $$\varepsilon_{\varepsilon_0}$$

$$\{\{0,1\},1\}$$ has growth rate $$\zeta_0$$

$$\{0,1\}\{\{0,1\},1\}$$ has growth rate $$\varepsilon_{\zeta_0 + 1}$$

$$\{1,1\}\{\{0,1\},1\}$$ has growth rate $$\varepsilon_{\zeta_0 + \omega}$$

$$\{\{\{\{0,1\},1\}\},1\}\{\{0,1\},1\}$$ has growth rate $$\varepsilon_{\zeta_0 2}$$

$$\{1\{\{\{0,1\},1\}\},1\}\{\{0,1\},1\}$$ has growth rate $$\varepsilon_{\zeta_0 \omega}$$

$$\{\{\{\{0,1\},1\}\}\{\{\{0,1\},1\}\},1\}\{\{0,1\},1\}$$ has growth rate $$\varepsilon_{\zeta_0^2}$$

$$\{\{1\{\{0,1\},1\}\},1\}\{\{0,1\},1\}$$ has growth rate $$\varepsilon_{\zeta_0^\omega}$$

$$\{\{\{\{0,1\}\}\{\{0,1\},1\}\},1\}\{\{0,1\},1\}$$ has growth rate $$\varepsilon_{\zeta_0^{\varepsilon_0}}$$

$$\{\{\{\{\{0,1\},1\}\}\{\{0,1\},1\}\},1\}\{\{0,1\},1\}$$ has growth rate $$\varepsilon_{\zeta_0^{\zeta_0}}$$

$$\{\{\{0,1\}\{\{0,1\},1\}\},1\}\{\{0,1\},1\}$$ has growth rate $$\varepsilon_{\varepsilon_{\zeta_0 + 1}}$$

$$\{\{0,1\},1\}\{\{0,1\},1\}$$ has growth rate $$\zeta_1$$

$$\{1\{0,1\},1\}$$ has growth rate $$\zeta_\omega$$

$$\{\{\{0,1\}\}\{0,1\},1\}$$ has growth rate $$\zeta_{\varepsilon_0}$$

$$\{\{\{\{0,1\},1\}\}\{0,1\},1\}$$ has growth rate $$\zeta_{\zeta_0}$$

$$\{\{0,1\}\{0,1\},1\}$$ has growth rate $$\eta_0$$

$$\{\{1,1\},1\}$$ has growth rate $$\varphi(\omega,0)$$

$$\{\{\{\{0,1\}\},1\},1\}$$ has growth rate $$\varphi(\varepsilon_0,0)$$

$$\{\{\{\{\{0,1\},1\}\},1\}$$ has growth rate $$\varphi(\zeta_0,0)$$

$$\{\{\{0,1\},1\},1\}$$ has growth rate $$\Gamma_0$$

See the pattern? $$\{n,m\}$$ is same at $$[[n]_m]$$, because growth rate is same.

$$\{0,0,1\}$$ has growth rate $$\psi(\psi_I(0))$$

$$\{0,n,1\}$$ has growth rate $$\psi(\psi_{I + n}(0))$$

$$\{0,0,2\}$$ has growth rate $$\psi(\psi_{I_2}(0))$$

$$\{0,0,n\}$$ has growth rate $$\psi(\psi_{I_n}(0))$$

$$\{0,0,\{0,0,1\}\}$$ has growth rate $$\psi(\psi_{I_I}(0))$$

$$\{0,0,0,1\}$$ has growth rate $$\psi(\psi_{I(1,0)}(0))$$

$$\{0,x,y,z\}$$ has growth rate $$\psi(\psi_{I_{I(1,z - 1) + y} + x}(0))$$

$$\{0,0,0,0,1\}$$ has growth rate $$\psi(\psi_{I(2,0)}(0))$$

$$\{0,x,y,z,w\}$$ has growth rate $$\psi(\psi_{I_{I(1,I(2,w - 1) + z) + y} + x}(0))$$

$$\{0,0,0,0,0,1\}$$ has growth rate $$\psi(\psi_{I(3,0)}(0))$$

$$\{0,0...0,0,1\}$$ (n zeroes) has growth rate $$\psi(\psi_{I(n-2,0)}(0))$$

$$\{0\{1*\}1\}$$ has growth rate $$\psi(\psi_{I(\omega,0)}(0))$$

$$\{0,1\{1*\}1\}$$ has growth rate $$\psi(\psi_{I(\omega,0)+1}(0))$$

$$\{0,0...0,0,1\{1*\}1\}$$ (n zeroes) has growth rate $$\psi(\psi_{I(n-2,I(\omega,0)+1)}(0))$$

$$\{0\{1*\}n\}$$ has growth rate $$\psi(\psi_{I(\omega,n-1)}(0))$$

$$\{0\{1*\}0,0...0,0,1\}$$ (n zeroes) has growth rate $$\psi(\psi_{I(\omega + n,0)}(0))$$

$$\{0\{1*\}0\{1*\}1\}$$ has growth rate $$\psi(\psi_{I(\omega 2,0)}(0))$$

$$\{0\{1*\}0...0\{1*\}0\{1*\}1\}$$ (n zeroes) has growth rate $$\psi(\psi_{I(\omega n,0)}(0))$$

$$\{0\{n*\}1\}$$ has growth rate $$\psi(\psi_{I(\omega^n,0)}(0))$$

$$\{0\{\{0,1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(\Omega,0)}(0))$$

$$\{0\{\{0,0...0,0,1\}*\}1\}$$ (n zeroes) has growth rate $$\psi(\psi_{I(I(n-2,0),0)}(0))$$

$$\{0\{\{0\{1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(I(\omega,0),0)}(0))$$

$$\{0\{... \{0\{1*\}1\} ...*\}1\}$$ (n nested) has growth rate $$\psi(\psi_{I(...I(\omega,0)...,0)}(0))$$ (n nested)

$$\{0\{0,1*\}1\}$$ has growth rate $$\psi(\psi_{I(1,0,0)}(0))$$

$$\{0\{0,1*\}0,1\}$$ has growth rate $$\psi(\psi_{I(1,1,0)}(0))$$

$$\{0\{0,1*\}0\{1*\}1\}$$ has growth rate $$\psi(\psi_{I(1,\omega,0)}(0))$$

$$\{0\{0,1*\}0\{0,1*\}1\}$$ has growth rate $$\psi(\psi_{I(2,0,0)}(0))$$

$$\{0\{1\{0\{0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(\omega,0,0)}(0))$$

$$\{0\{\{0\{\{0\{0,1*\}1\}*\}1\}\{0\{0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(I(1,0,0),0,0)}(0))$$

$$\{0\{\{0\{0,1*\}1\}\{0\{0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(1,0,0,0)}(0))$$

$$\{0\{\{1\{0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(M^\omega,0)}(0))$$

$$\{0\{\{\{0,1\}\{0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(M^\Omega,0)}(0))$$

$$\{0\{\{\{0\{0,1*\}1\}\{0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(M^M,0)}(0))$$

$$\{0\{\{0,1\{0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(\varepsilon_{M+1},0)}(0))$$

$$\{0\{\{0,0,1\{0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(\psi_{I_{M+1}}(0),0)}(0))$$

$$\{0\{\{0\{0,1*\}2\}*\}1\}$$ has growth rate $$\psi(\psi_{I(M_2,0)}(0))$$

$$\{0\{\{0\{0,1*\}0,1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(M(1,0),0)}(0))$$

$$\{0\{\{0\{0,1*\}0\{1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(M(\omega,0),0)}(0))$$

$$\{0\{\{0\{0,1*\}0\{0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(\Xi(3,0),0)}(0))$$

$$\{0\{\{0\{1,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(\Xi(\omega,0),0)}(0))$$

$$\{0\{\{0\{0,2*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{I(\Xi(K,0),0)}(0))$$

$$\{0\{\{0\{0,2*\}0\{0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{\Omega(1,1,0,0)}(0))$$

$$\{0\{\{0\{0,2*\}0\{0,2*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{\Omega(2,0,0,0)}(0))$$

$$\{0\{\{0\{0,3*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{\Omega(1,0,0,0,0)}(0))$$

$$\{0\{\{0\{0,n*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{\Omega(\Omega\{2\}^{n+1})}(0))$$

$$\{0\{\{0\{0,\{0\}*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{\Omega(\Omega\{2\}^\omega)}(0))$$

$$\{0\{\{0\{0,0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{\Omega(\Omega\{2\}^{\Omega\{2\}})}(0))$$

$$\{0\{\{0\{0,0,1*\}0,1\}*\}1\}$$ has growth rate $$\psi(\psi_{\Omega(\Omega\{2\}(1,0))}(0))$$

$$\{0\{\{0\{0,0,1*\}0\{0,0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{\Omega(\Omega\{2\}(\Omega\{3\}^{\Omega\{3\}}))}(0))$$

$$\{0\{\{0\{1,0,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{\Omega(\Omega\{\omega\})}(0))$$

$$\{0\{\{0\{0,1,1*\}1\}*\}1\}$$ has growth rate $$\psi(\psi_{\Omega(\Omega\{0\}\{1\})}(0))$$

More coming soon!