  We explore the possibility that our universe's current accelerated expansion
is explained by a quintessence model with an exponential scalar potential, $V
=V_0\, e^{-\lambda\, \phi}$, keeping an eye towards $\lambda \geq \sqrt{2}$ and
an open universe, favorable to a string theory realisation and with no
cosmological horizon. We work out the full cosmology of the model, including
matter, radiation, and optionally negative spatial curvature, for all
$\lambda>0$, performing an extensive analysis of the dynamical system and its
phase space. The minimal physical requirements of a past epoch of radiation
domination and an accelerated expansion today lead to an upper bound $\lambda
\lesssim \sqrt{3}$, which is driven slightly up in the presence of
observationally allowed spatial curvature. Cosmological solutions start
universally in a kination epoch, go through radiation and matter dominated
phases and enter an epoch of acceleration, which is only transient for
$\lambda>\sqrt{2}$. Field distances traversed between BBN and today are
sub-Planckian. We discuss possible string theory origins and phenomenological
challenges, such as time variation of fundamental constants. We provide
theoretical predictions for the model parameters to be fitted to data, most
notably the varying dark energy equation of state parameter, in light of recent
results from DES-Y5 and DESI.
