With the rumored Gobuster having animal-inspired partners, one is to think there will be furry. Furriness and animal friends aren't that uncommon.
Doggy Krugger (Dekaranger - 2004)
Literally furry, a dog alien from Anubis, he was the boss of the Dekaranger and became DekaMaster. He had a long snout and was once asked how it fit in his helmet, he told them not to ask, something of that effect. He had a hinted romance with Swan. He appeared in Gokaiger earlier this year, reuniting his suit actor Hideaki Kusaka and voice actor Tetsu Inada.
Doggie Cruger (SPD -2005)
A more reptilian-look of a dog alien compared to his predecessor/counterpart, Doggie was more angry and a bit childish for my taste. He lost his world and was missing his wife.
Smoky/Jenji (Magiranger - 2005 /Mystic Force - 2006)
Both pals of MagiShine/Solaris Knight, he was a genie cat who had liked getting in trouble. Smoky was stuck to the lamp and could only be with legs for a short while. They both had a permanant grin and were part of the lamp attack their master did.
Master Shafu (Gekiranger - 2007)
Literally furry, he is a lynx, he was once human and in love with Rageku. He was Gekiranger's mentor and often gave him riddles and had them do menial work to help them develop their training to fulfill tasks. He often rang his triangle. Master Mao of Jungle Fury became him in the last episode. Master Shafu made a brief cameo in episode 7 of Gokaiger.
Master Elehan Kimpo (Gekiranger - 2007)
Pervert and trained Ran/GekiYellow. Jungle Fury's Master Phant took this form once.
Master Bat Li (Gekiranger - 2007)
Name based on Jet Li, he was serious, he trained Retsu/GekiBlue and the blind Master Swoop took this form once.
Master Sharkie Chan (Gekiranger - 2007)
He reminds me of Kung Fu Panda as he was like a kid, never had a trainee before. He trained Jan/Geki Red and Master Finn---who was more experience and RJ's dad, took this form at end of PRJF.
Master Gorie Yen (Gekiranger - 2007)
He was serious and wrote books and lived in New York, he trained Jan and challenged him. His JF counterpart took his form once.
Master Pyon Biao
His Spirit World counterpart took his form once, there wasn't much to his character in Gekiranger except that he once dressed up for Christmas and was mistaken for a Reindeer.
Master Michelle PengShe lived in Sweden and taught Gou/GekiViolet how to do a legendary move. Her counterpart took her form in Jungle Fury.