Ly-Chee

 was a human male of somewhat advanced years with whom the Seventh Doctor travelled for an indeterminate time.

Known biographical details
On the planet Tora, somewhere in the Crab Nebula, the companion-less Seventh Doctor was once walking close to a mountain named Kabuki. Suddenly, he was beset by vandals, and Ly-Chee, a hermit who lived on the mountain, came down swiftly on the attackers. Though the Doctor thanked Ly-Chee for his able assistance, the Doctor's knowledge of Venusian aikido proved sufficient to his own defence.

Ly-Chee then offered the Doctor a respite of nettle tea back at his home in a mountain cave. Over the drink, Ly-Chee explained that he had been occupied all his contemplative life — over 30 Toran years — with the search for the Holy Number of Nirvana — a mathematical puzzle.

The Doctor's mug of tea turned into a few days of study. Eventually, though, the Doctor deduced that the answer was 7, thereby unceremoniously ending Ly-Chee's life work. Desperate for a new challenge, Ly-Chee asked the Doctor if he had a puzzle that needed solving. The Doctor quickly came up with a human classic. "Why", the Doctor asked, "did the chicken cross the road?"

The question was one that had never been posed to the hermit. So he put himself in deep seclusion. For an entire week, he plunged himself under a waterfall to think. Evntually, the Doctor broke the wise man's deliberations, unable to let him waste any more of his life with what was essentially just a joke. "to get to the other side," came the predictable answer. Ly-Chee was so crestfallen at having once again encountering a riddle with an incredibly simple answer that he purposed to return to town and look for a "real" job. Guilty at having essentially ruined his friend's life, the Doctor offered to give him a lift to town in the TARDIS.

The two then travelled together for a number of adventures before the Doctor eventually found the town to which Ly-Chee had originally wanted to go. Happy to be back in a place with which he was familiar, Ly-Chee immediately headed towards the nearest pub to order himself a stiff drink. Only then, after an unknown but apparently considerable amount of time, did the two companions part each other's company. (IHP: ''The Enlightenment of Ly-Chee the Wise')

Behind the scenes

 * Ly-Chee is a character whose companion status is indisputable, but whose journeys with the Doctor are quite probably the most compressed of anyone in all of Doctor Who fiction. His stint with the Doctor is compressed into just one completely wordless panel — albeit an oversized one.  Nevertheless, that panel does convey a sense of the enormity of his experience with the Doctor.
 * The narration boxes give the character the longer names of "Ly-Chee the Wise", "Ly-Chee the Considerate", and "Ly-Chee the Human Being", but it's not clear these are names that other people within the narrative would have actually used for him. These seem to be tongue-in-cheek appellations that give the narrator an appropriately ironic tone.