This whole "doubles" thing should almost have a thread of its own, but.....
The worst "doubles/lookalike" plots (and a sure sign that a show has jumped the shark and is running out of ideas) are in the genre "Character Meets Real-Life Star Who Plays Him/Her." The two that come to mind are Lucy Carter meeting Lucille Ball on
Here's Lucy, and Fred Sanford meeting Redd Foxx on
Sanford and Son. I'm sure there are a few other examples that could be cited.
Also, episodes where actors play multiple roles were traditionally limited by logistics and technology: the basic split-screen technique was stilted and contrived, locking scenes involving the two doppelgangers into a standardized, static "opposite sides of the screen in profile and always at least two feet apart" configuration. (Spelled only occasionally by the equally fakey "over the shoulder shot with the back of an actor in the foreground who only vaguely resembles the star in height and appearance" variation.) :

These limitations, and the sometimes poor staging of the effect, added an element of contrivance that rendered an already hackneyed idea even more ludicrous. Plus, not every actor, no matter how talented, is able to convincingly master the art of playing a scene "with themselves."
But digital technology has changed that. The first time I was "wowed" by such technology in a TV show was the episode of
Family Matters in which Urkel clones himself. The two Steves were able to cross in front of each other (even do a 360, circling around each other), embrace or otherwise physically interact, etc., and the effect is flawless. Coupled with the talent of Jaleel White, you could
really believe that there were two Urkels interacting with each other. The realism of the illusion greatly improves the entertainment value of what could well have been another typical "doubles" plot in the classic mode.
Stargate SG-1 frequently used (some would say overused) the technique, and there was usually at least one episode a season in which the characters encountered doubles in some fashion or other. Fortunately, the excellent writing and the consummate talent of the actors helps to pull it off. There is a scene that is an absolute scream in a late-season episode involving a rift that has multiple parallel universes "leaking" into ours, in which about 20-odd Colonel Carters are all in the same room together, believably talking to each other, moving around, etc. (Leading to a funny moment when General Landry enters the room and calls Sam's name, and all the Sams react simultaneously, prompting him to chuckle and clarify that he meant "
MY Colonel Carter.") One can only imagine how much time and effort -- carefully staging Amanda Tapping's position, movements, and gestures in 20+ takes, not to mention the amount of post-production digital magic -- must have been expended on a scene that lasts perhaps a minute on screen!