There currently isn't any method of controlling where offset text for an axis gets placed. This causes problems when changing tick locations (e.g. tick_top and tick_right).

Axis.offset_text_position appears to have been intended to be the control, but changing ax.xaxis.offset_text_position and ax.yaxis.offset_text_position has no effect. Furthermore, there doesn't appear to be a code path where offset_text_position is used, or any other mechanism for changing the offset text's position. XAxis._update_offset_text_position (and the equivalent in YAxis) will need to be updated to take this into account.

Consider the following example:

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0, 1e8)
y = np.exp(x**0.2)

fig, ax = plt.subplots()
ax.plot(x, y)

ax.xaxis.tick_top()
ax.yaxis.tick_right()

plt.show()

In this case, the 1e17 corresponds to the y-axis, while the 1e8 corresponds to the x-axis. Ideally, these should be aligned so that it's clear which axis they correspond to.

For the y-axis, we can move the offset text to the left-hand side, but it will still be in a position that overlaps with the x-axis's ticklabels. Regardless, there's no corresponding method for the x-axis. As an example:

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0, 1e8)
y = np.exp(x**0.2)

fig, ax = plt.subplots()
ax.plot(x, y)

ax.xaxis.tick_top()
ax.yaxis.tick_right()
ax.yaxis.set_offset_position("right")

plt.show()

Note: Originally noticed by user3160855 in a Stackoverflow question.