According to the official GPS routes posted here on HAZ, the elevations can vary depending on how you hike the trail.
If you through-hike going northbound, the AEG is 113,088 feet.
If you through-hike going southbound, the AEG is 113,537 feet.
(I was surprised by how similar they are!)
Many people section hike the trail rather than through-hiking it, and don't always hike each segment in the same direction. There are 25 sections that have a greater AEG going southbound, and 18 whose AEG is higher going northbound.
If you choose the higher AEG for each segment, the total would be 139,456 feet.
If you prefer to hike each segment in the predominantly "downhill" direction, the total AEG would be 87,169 feet.
The numbers would vary if you start and end at points that are not at the officially designated spots between segments.
But if you hike in one direction you will gain ~113,000 feet. However, you can cover every step of the trail in as little as ~87,000 feet, or as much as ~140,000 feet of total accumulated gain depending on how you break it up and which direction you hike the various segments.