>>> from scipy.special import ellipkinc
>>> ellipkinc(0.9272952180016123, 0.68359375000000011)
nan
>>> ellipkinc(0.9272952180016123, 0.6835937500000002)
2.0518660200390668
>>> ellipkinc(0.9272952180016123, 0.68359375)
1.0259330100195332
>>> ellipkinc(0.9272952180016123, 0.68359374)
1.0259330081166262

In [91]: phi = 0.9272952180016123

In [92]: mbad = 0.68359375000000011

In [93]: m = np.nextafter(mbad, 0)

In [94]: mvals = []

In [95]: for j in range(10):
   ....:     mvals.append(m)
   ....:     m = np.nextafter(m, 1)
   ....:

In [96]: mvals
Out[96]:
[0.68359375,
 0.68359375000000011,
 0.68359375000000022,
 0.68359375000000033,
 0.68359375000000044,
 0.68359375000000056,
 0.68359375000000067,
 0.68359375000000078,
 0.68359375000000089,
 0.683593750000001]

In [97]: ellipkinc(phi, mvals)
Out[97]:
array([ 1.02593301,         nan,  2.05186602,  2.05186602,  2.05186602,
        2.05186602,  1.02593301,  1.02593301,  1.02593301,  1.02593301])