Companion animals: Bond creatures, telepathic or magical companion animals; think anything from Valdemar's horses and Pern's dragons to His Dark Material's daemonsa non-human with an incredibly powerful and meaningful link to a human.
Not high fantasy: I'm fine with second-world fantasy; it's high fantasy's typical tropes and archetypes that I dislike: elves, dwarves, magicians; made up languages, capitalized nouns; rambling epic quests that bridge twelve books.
...or series: Stand-alone novels within a series are fine, whether or not they benefit from knowing the rest of the seriesjust please point me towards those individual books, not the series as a whole (if they all stand alone, then point me towards the best one or perhaps the best starting place). A short-running, completed series (around three books) may work if you think it truly shouldn't be missed, but honestly I'd prefer to avoid those. Stand-alone books with no connection to series are my favorite, and I detest ongoing and/or sprawling epics.
Almost every companion animal book list I can find is a list of series, so this is me asking for individual booksif they come from series, fine, as long as I don't need to read all dozen books of them.
I have read some Valedmar (Heralds of Valdemar, The Last Herald-Mage), His Dark Materials, A Companion to Wolves (but not yet the sequel), Tepper's Grass, but have not actually read any Pern. I prefer almost any animal to dragons, but could give dragons a try; I like my books more dark, psychological, and id-driven rather than fluffy, although I get that this subgenre tends towards wish-fulfillment.