Sorry EMS, but I actually just got another response from the matplotlib mailling list (Thanks goes out to Benjamin Root).
The code I am looking for is adjusting the savefig call to:
fig.savefig('samplefigure', bbox_extra_artists=(lgd,), bbox_inches="tight")
#Note that the bbox_extra_artists must be an iterable
This is apparently similar to calling tight_layout, but instead you allow savefig to consider extra artists in the calculation. This did in fact resize the figure box as desired.
import matplotlib.pyplot as plt
import numpy as np
plt.gcf().clear()
x = np.arange(-2*np.pi, 2*np.pi, 0.1)
fig = plt.figure(1)
ax = fig.add_subplot(111)
ax.plot(x, np.sin(x), label="Sine")
ax.plot(x, np.cos(x), label="Cosine")
ax.plot(x, np.arctan(x), label="Inverse tan")
handles, labels = ax.get_legend_handles_labels()
lgd = ax.legend(handles, labels, loc="upper center", bbox_to_anchor=(0.5,-0.1))
text = ax.text(-0.2,1.05, "Aribitrary text", transform=ax.transAxes)
ax.set_title("Trigonometry")
ax.grid('on')
fig.savefig('samplefigure', bbox_extra_artists=(lgd,text), bbox_inches="tight")
This produces:
[edit] The intent of this question was to completely avoid the use of arbitrary coordinate placements of arbitrary text as was the traditional solution to these problems. Despite this, numerous edits recently have insisted on putting these in, often in ways that led to the code raising an error. I have now fixed the issues and tidied the arbitrary text to show how these are also considered within the bbox_extra_artists algorithm.