Boxwood
You may not have heard about this shrub under this name, but you have definitely seen it around many houses. They are shrubs that come in different varieties, with the most common ones coming from two main species: the common ones and the Japanese ones. They are popular in the dwarf variety for landscaping, though there are some species that can grow to bigger heights.
They are the perfect choice for your live fence as they can be left natural or sheared if you want to give them a specific shape, and they are amazing at keeping deer away as they are definitely unappealing to them. As deer do not like to eat this type of shrub, they end up being amazing at deterring them if they are the first thing they encounter on your property!