The original battery included in the GoTenna Mesh (GTM) V1.0 2017 (with RF Shield) is a 589mah 1S LiPo (Lithium Polymer).
LiPo batteries are identified by their composition, where (S) = Cells in Series and § = Cells in Parallel. In the GTM, the 1S means we have one cell in series for a 3.7V nominal (3.0V low and 4.2V high). If upgrading the battery the minimalist way, you will want to keep to a 1S (3.7V nominal) LiPo battery to ensure the charging circuit still works appropriately and you do not fry the other components on the PCB.
My first mod is to replace the on-board batt with an off-board higher capacity 1S LiPo battery using XT30 connectors to allow for an easy change-out of batteries. The batteries I’ll use will be 10,000 mAh 1S LiPo’s. These will generate 16.9 days worth of charge.
Instead of attempting a solar panel/charge controller/battery bank to charge the battery, my plan is to simply replace the battery and charge it at home.