Priest of Titania does not have summoning sickness.
There are at least four Elves on the battlefield.
Steps
Activate Priest of Titania by tapping it, adding ({G} magic symbol) for each Elf on the battlefield.
Activate Wirewood Lodge's second ability by paying ({G} magic symbol) and tapping it, targeting Priest of Titania.
Rings of Brighthearth triggers, causing you to pay ({2} magic symbol) to copy Wirewood Lodge's ability, targeting Priest of Titania.
Resolve the first Wirewood Lodge ability, untapping Priest of Titania.
Activate Priest of Titania by tapping it, adding ({G} magic symbol) for each Elf on the battlefield.
Resolve the second Wirewood Lodge ability, untapping Priest of Titania.
Activate Deserted Temple's second ability by paying ({1} magic symbol) and tapping it, targeting itself.
Rings of Brighthearth triggers, causing you to pay ({2} magic symbol) to copy Deserted Temple's ability, targeting Wirewood Lodge.
Resolve both Deserted Temple abilities, untapping it and Wirewood Lodge.
Repeat.
Once you have infinite green mana, you may repeat steps 7 through 9 targeting any other land you control instead, or repeat the combo targeting any Elves you control.