  The electric Weak Gravity Conjecture demands that axions with large decay
constant $f$ couple to light instantons. The resulting large instantonic
corrections pose problems for natural inflation. We explore an alternative
argument based on the magnetic Weak Gravity Conjecture for axions, which we try
to make more precise. Roughly speaking, it demands that the minimally charged
string coupled to the dual 2-form-field exists in the effective theory. Most
naively, such large-$f$ strings curve space too much to exist as static
solutions, thus ruling out large-$f$ axions. More conservatively, one might
allow non-static string solutions to play the role of the required charged
objects. In this case, topological inflation would save the superplanckian
axion. Furthermore, a large-$f$ axion may appear in the low-energy effective
theory based on two subplanckian axions in the UV. The resulting effective
string is a composite object built from several fundamental strings and domain
walls. It may or may not satisfy the magnetic Weak Gravity Conjecture depending
on how strictly the latter is interpreted and on the cosmological dynamics of
this composite object, which remain to be fully understood. Finally, we recall
that large-field brane inflation is naively possible in the codimension-one
case. We show how string-theoretic back-reaction closes this apparent loophole
of large-$f$ (non-periodic) pseudo-axions.
