For several years I lived on an island nature reserve with a large population of hares. It was a delight to see them loping around perfectly at ease. We have moved, but still have one that regularly visits the garden. One of my colleagues also works for the Hare Preservation Trust (I'm in the UK).
For me, it would be the toad - Bufo bufo. I have enjoyed their presence from childhood, but looked after an area that was home to a large breeding colony at one point in my career. I improved the water levels in the wet meadows at one point which resulted in the colony expanding. They had to cross a road to reach the meadows, so I started up a volunteer toad patrol in the mornings and evenings during breeding seasons to prevent them from getting run over. It was initially just me and a couple of other people that I met out at that time of the morning etc, but soon grew.
This became the largest recorded road-crossing toad colony in the UK at one point - and was covered by the media briefly. The data that we recorded then proved critical to overturning a housing development on the site and led to it being designated as a nature reserve. I consider this to be a highlight of my career.
There is belief that they have a precious stone in their heads - and looking into their amber eyes it is easy to see why.
Bats, stag beetles, adders, fallow deer and dragonflies are others that have a similar effect on me: each prompting that deeper experience of connection.