Quadratic-inflation expansion from Planck density ($N\simeq 4\pi (m_P/m{)}^2$)

For $V=m^2{\varphi }^2/2$ and unreduced Planck mass, the slow-roll e-fold count is $N=2\pi ({\varphi }_i^2-{\varphi }_f^2)/m_P^2$. A formal initial potential $V_i\sim m_P^4$ gives ${\varphi }_i^2\sim 2m_P^4/m^2$, hence the displayed large-field estimate and expansion factor $e^N$. The final-field contribution is order one in $N$. Without specifying $m/m_P$ there is no unique numerical expansion factor. The extrapolation to Planck density is a classical estimate, rather than a controlled treatment of quantum-gravity corrections.