The game clock is used to prevent players from overly delaying the game (the keywords being "overly delaying the game"). Are there times in a timed game that stalling one's own development of the board makes sense, and if so, what would be the best example of this taking place?
In the bughouse variant a situation can definitely arise in which it makes sense to stall for time. If I'm facing checkmate next move on my board no matter what I do, then if my partner's opponent has less time on the clock than I do, I should just sit tight and hope that my partner can win on his board before my time runs out. (And if my partner's opponent has more time than I do in that situation, then he should just sit tight as well and force me to either accept mate or flag first.) But I realize you are actually asking about chess proper here, not bughouse.
I can't think of any scenario in which it makes objective sense, in terms of the game play, to stall for time, assuming you are talking about a situation in which: I already know what my next move is going to be (and have no intention of reviewing my decision any further), but I let my clock tick down instead of making my move immediately. But a player might have a psychological reason to stall at times.
In this interview (with accompanying videos in Russian) of grandmaster and noted commentator Sergey Shipov, conducted just recently during the ongoing world championship match, the topic arises as to the amount of time top players like Gelfand, Kramnik and Kasparov spend when making moves in positions that they've already analyzed during their home preparations. (See section "6. Kasparov puts on a show.") Shipov opines:
For example, when, in Kazan, Gelfand thought for almost 40 minutes about one continuation in that game against Grischuk it later became clear that he’d looked at it all before. [17. Rd1 in game 6 of the candidates' final] ... he was considering some nuances, subtleties. He’d looked at the plan itself, but perhaps he was clarifying some of the concrete details. That depends greatly on the chess player. Kramnik won’t, for example, think for 20 minutes, never mind 40 minutes, about something he’s already looked at. He’ll play quickly. Kasparov, on the other hand, might do that. With Kasparov a position could be absolutely familiar to him but he’d think for 20 minutes, look at it, make various gestures, and it seemed to everyone that Kasparov was creating and coming up with something at the board. In actual fact, however, it was a one-man performance based around the creative process. (My emphasis)
Shipov doesn't indicate exactly what he thinks Kasparov's underlying motivation for this was supposed to be, but he is clearly suggesting that Kasparov wanted either his opponents or his spectators not to be sure when he was in his preparation and when he was extemporizing. So that's one possible answer.
Ed Dean has an excellent answer about most cases (and even touches on what I'm discussing), but there is one situation that I can think of where stalling is an excellent idea:
This mostly only applies to stronger players (2300+), but the information is good...
Sometimes it is very useful to spend a lot of time on an early opening move to try to confuse the opponent. If the opponent thinks that you don't know the position, they are more likely to play natural looking moves. This is frequently a bad idea, especially in complicated positions where natural looking moves fail to various tactics.
The idea behind waiting a long time before moving is to lure the opponent into making those natural looking moves, or to think that the opponent's position is good when really the position has been analyzed fully and is deemed to be good for the player. Basically, the point is that the opponent won't be fully on the lookout for sharp tactics going forward, especially in double edged positions.
Overall, I don't see any reason to do this unless you want to annoy your opponent.
If your opponent is advanced, I think you would just be wasting time, especially if you think your opponent is going to be psychologically tricked into making some sort of blunder.
If your opponent is a beginner, this may have some psychological effect as it your opponent may think that because you took so long to move, you had to have calculated some long winning combination and your opponent may start to panic and look for what you saw.
Regarding the stalling with a beginner, I know this is true, because it works on me. If my opponent takes a while to move, I usually think they have something up there sleeve unless it is some horrible mistake like hanging their queen.
There was a game where Korchnoi was completely lost and deliberately ran himself into time trouble so that the opponent would try to take advantage of it rather than playing the merits of the position. I believe it worked and he turned the game around.
On a personal note, I once found myself facing an 8-year old who was a superior player; I knew he was a superior player before we started, having watched several of his other games at the club downtown.
I intentionally let extra time go off my clock on every move, even when it was something blatantly obvious. Relatively quickly in the game, I found myself down a minor piece... but it became clear by mid-game that he didn't have the mental stamina to keep it up: he started fidgeting, and his attention started wandering. Eventually, he made a blunder I was able to capitalize on, and I managed to win.
I spoke to his father afterward - and he related that delaying the game until his son's mental fortitude wore off was the only way he could win, too. With that said, I'm sure that the young man - whom I have since lost touch with - became a MUCH better player once he matured enough to sustain a longer play style.
In team matches it can sometimes make some kind of sense. One case I saw was white was miles up on time and in a somewhat better position, but certainly not clearly winning. At that point Black offered a draw, to which white responded by waiting for the other boards to become more clear before he decided whether the team needed a win from him, or if a draw was sufficient. In the end neither was (I was on the side of Black)!