We present the complete set of {it Hubble Space Telescope} imaging of the binary neutron star merger GW170817 and its optical counterpart AT 2017gfo. Including deep template imaging in F814W, F110W, F140W, and F160W at 3.4 years post-merger, we re-analyze the full light curve of AT 2017gfo across 12 bands from 5--1273 rest-frame days after merger. We obtain four new detections of the short $gamma$-ray burst (GRB) 170817A afterglow from 109--170 rest-frame days post-merger. These detections are consistent with the previously observed $beta=-0.6$ spectral index in the afterglow light curve with no evidence for spectral evolution. We also analyze our limits in the case of novel late-time optical and IR emission signatures, such as a kilonova afterglow or infrared dust echo, but find our limits are not constraining in these contexts. We use the new data to construct deep optical and infrared stacks, reaching limits of $M=-6.3$ to $-4.6$ mag, to analyze the local environment around AT 2017gfo and low surface brightness features in its host galaxy NGC 4993. We rule out the presence of any globular cluster at the position of AT 2017gfo to $2.3 times 10^{4} L_{odot}$, including those with the reddest $V-H$ colors. Finally, we analyze the substructure of NGC 4993 in deep residual imaging, and find shell features which extend up to 71.8arcsec (14.2 kpc) from the center of the galaxy. We find that the shells have a cumulative stellar mass of $6.3times10^{8} M_{odot}$, roughly 2% the total stellar mass of NGC 4993, and mass-weighted ages of $>$3 Gyr. We conclude that it was unlikely the GW170817 progenitor system formed in the galaxy merger, which based on dynamical signatures and the stellar population in the shells mostly likely occurred 220--685 Myr ago.
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