  We present deep upper limits from the 2014 Murchison Widefield Array (MWA)
Phase I observing season, with a particular emphasis on identifying the
spectral fingerprints of extremely faint radio frequency interference (RFI)
contamination in the 21~cm power spectra (PS). After meticulous RFI excision
involving a combination of the \textsc{SSINS} RFI flagger and a series of
PS-based jackknife tests, our lowest upper limit on the Epoch of Reionization
(EoR) 21~cm PS signal is $\Delta^2 \leq 1.61\cdot10^4 \text{ mK}^2$ at
$k=0.258\text{ h Mpc}^{-1}$ at a redshift of 7.1 using 14.7 hours of data. By
leveraging our understanding of how even fainter RFI is likely to contaminate
the EoR PS, we are able to identify ultra-faint RFI signals in the cylindrical
PS. Surprisingly this signature is most obvious in PS formed with less than an
hour of data, but is potentially subdominant to other systematics in
multiple-hour integrations. Since the total RFI budget in a PS detection is
quite strict, this nontrivial integration behavior suggests a need to more
realistically model coherently integrated ultra-faint RFI in PS measurements so
that its potential contribution to a future detection can be diagnosed.
