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First you do NOT want to mount the battery inside a closed area, as a charging battery emits some gases (explosive type). A well ventilated area, like on the tongue is a good place. You can make an inexpensive battery holder using an old bed frame. Cut it to the size you need and have someone weld it to the tongue. HarborFreight also has an inexpensive tongue mount storage container that regular batteries will fit in.
The question regarding the size battery you need, depends a lot on what you plan on using your TT for. If you are only going to go to CG's that have full service you will not need more than an 100 AH battery, the standard "Marine" type battery will do just fine. If you plan on a lot of dry-camping, you would want a larger capacity "TRUE" deep cycle battery. There are a lot of posts that cover batteries on this site.
If you are looking at dry camping, you will also want to replace those energy hungry bulbs in the TT with LEDS. It is amazing how much energy regular bulbs use.
2013 Jayco Eagle 284BHS
250Watt Grape Solar Panel, MorningStar MPPT 60 Charge Controller
1500 Watt Ramsond PSI, 2 Trojan T145 Batteries (260Ah)
2 - AirSight Wireless IP Cameras (used as rear view cameras)
EnGenius WI-FI extender, D-Link wireless (n) modem
MagicJack Internet Phone
2012 Ford F150XLT, EcoBoost w/3.73
157" Wheel base, HD Towing Package
Our Solar Album http://www.jaycoowners.com/album.php?albumid=329