Doctor Who: Most Useless Companion

[Slappy]

Mel6 & 7:  Little more than a shreiking Harpy in constant trouble.
Perpugilliam Brown(Peri)5&6 If it wasn't for her looks, and her rack, she would be unwatchable. Leela was there for sex too, but at least she could handle herself.
Katarina 1: Helped carry Steven only to die a few episodes later.  Almost as short a tenure as Adam Mitchell (9th Doctor).  Just as ignorant as Leela but dumber than a rock.
Nyssa 4&5:Useless until she was amazingly for the first time a Healer in Terminus. (When she left)
Dodo 1:If the name doesn't say it.  Her contract wasn't renewed and just disappeared after Wargames episode 2.  In the final episode, off screen she decided not to rejoin the Tardis crew

[The Mighty King Cobra]

Kamelion, hands down.  Completely useless, moreso even than Katarina.

[Slappy]

I did consider Kam, but while from a technical aspect he was a "big budget" disaster, from a story point he was important on more than one occassion.