What is the name given to the ordinal $\omega\uparrow^\omega\omega$?

Question

In this MathOverflow question, the ordinal $\omega\uparrow^\omega\omega$ is defined. It seems like a quite natural ordinal to me, since it is of course \begin{equation} \begin{array}{*8{>{\displaystyle}c}} & \omega \uparrow^\omega \omega \\ = & \sup \{ & \omega \uparrow^0 2, & \omega \uparrow^1 2, & \omega \uparrow^2 2, & \omega \uparrow^3 2, & \ldots & \} \\ = & \sup \{ & \omega \cdot 2 = \omega + \omega, & \omega^2 = \omega \cdot \omega, & ^2 \omega = \omega^\omega, & ^\omega\omega = \omega^{\omega^{\omega^{\cdot^{\cdot^\cdot}}}} & \ldots & \} \\ \end{array} \end{equation} So in a sense it is the result of saying "and so on" to the game of always getting to the next hyperoperation by saying "and so on".
What is the name of this ordinal?