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Quadritetrep is equal to $$f_{\varepsilon_0\uparrow\uparrow4}(10)$$ using the fast-growing hierarchy.[1] The term was coined by wiki user Denis Maksudov.

It is equal to $$f_{\omega^{\omega^{\omega^{\omega^{\varepsilon_02}}}}}(10)=f_{\varphi_0(\varphi_0(\varphi_0(\varphi_0(\varphi_1(0)+\varphi_1(0)))))}(10)$$ using Veblen function $$\varphi_\alpha(\beta)$$.

Etymology

The 3 parts of the name, "quadri", "tetr" and "ep", mean 4, tetration and $$\varepsilon_0$$ respectively, which form $$\varepsilon_0\uparrow\uparrow4$$ when concatenated backwards. So the full name indicates the ordinal index of the FGH.

Approximations in other notations

Notation Approximation
BEAF $$\{10,10((X \uparrow\uparrow X)^4) 2\}$$[2]
Extended Cascading-E Notation $$E10(\#\text{^^}\#)\text{^}(\#\text{^^}\#)\text{^}(\#\text{^^}\#)\text{^}(\#\text{^^}\#)10$$
Hardy hierarchy $$H_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}}(10)$$
Slow-growing hierarchy $$g_{\vartheta(\varepsilon_{\Omega 2}^{\varepsilon_{\Omega + 1}^{\varepsilon_{\Omega + 1}^{\varepsilon_{\Omega + 1}}}})}(10)$$

Sources

1. My system of number names (FGS) - Traveling To The Infinity
2. Using particular notation $$\{a,b (A) 2\} = A\ \&\ a$$ with prime b.
Numbers by Denis Maksudov
Alpha series
Epsilon series
Epsilon(0)-multiplication series: Bultep · Trultep · Quadrultep · Quintultep · Sextultep · Septultep · Octultep · Nonultep · Dekultep · Hektultep · Kilultep · Megultep · Gigultep · Terultep · Petultep · Exultep · Zettultep · Yottultep
Epsilon(0)-exponentiation series: Bexep · Trexep · Quadrexep · Quintexep · Sextexep · Septexep · Octexep · Nonexep · Dekexep · Hektexep · Kilexep · Megexep · Gigexep · Terexep · Petexep · Exexep · Zettexep · Yottexep
Epsilon(0)-tetration series: Bitetrep · Tritetrep · Quadritetrep · Quintitetrep · Sextitetrep · Septitetrep · Octitetrep · Nonitetrep · Dekotetrep · Hektotetrep · Kilotetrep · Megotetrep · Gigotetrep · Terotetrep · Petotetrep · Exotetrep · Zettotetrep · Yottotetrep
Inserted epsilon series: Uninep · Binep · Trinep · Quadrinep · Quintinep · Sextinep · Septinep · Octinep · Noninep · Dekinep · Hektinep · Kilinep · Meginep · Giginep · Terinep · Petinep · Exinep · Zettinep · Yottinep
Inserted zeta series
Inserted eta series
Phi series
Inserted gamma series
Theta series
Psi series
$$I_\alpha$$ series
$$I(\alpha,\beta)$$ series
Initial $$I(\alpha,\beta)$$ series: Unimah · Bimah · Trimah · Quadrimah · Quintimah · Sextimah · Septimah · Octimah · Nonimah · Dekimah · Hektimah · Kilimah · Megimah · Gigimah · Terimah · Petimah · Eximah · Zettimah · Yottimah
Inserted $$I(\alpha,\beta)$$ series: Uninimah · Binimah · Trinimah · Quadrinimah · Quintinimah · Sextinimah · Septinimah · Octinimah · Noninimah · Dekinimah · Hektinimah · Kilinimah · Meginimah · Giginimah · Terinimah · Petinimah · Exinimah · Zettinimah · Yottinimah
M-series

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