A player's character is in possession of a bomb wired up to a clock. If that character cast time stop, would the bomb keep running while time stop is in effect or would the timer stop? Or does the clock only stop after the bomb is no longer in the character’s possession?
The rules are unclear. The only thing time stop has to say on the subject of carried items is
You cannot move or harm items held, carried, or worn by a creature stuck in normal time, but you can affect any item that is not in another creature’s possession.
“Can” implies ability but not requirement. It implies that you can also choose not to. But “affect” in this context is about your actions during the time stop effect, whether you can pick up, move, or cast spells on those objects, not whether or not time stop itself affects them. And it doesn’t address items carried by the caster at all.
In fact, the initial description of time stop is
This spell seems to make time cease to flow for everyone but you. In fact, you speed up so greatly that all other creatures seem frozen, though they are actually still moving at their normal speeds.
The use of “everyone” and “creatures” seems to imply that non-creature objects are not affected. But in reality, the spell affects you, not any creature or object. It makes no sense for you to be moving super-fast, but somehow everything that doesn’t count as a creature does too.
Certainly, your possessions seem to be moving as fast as you—the spell doesn’t mention them pinning you in place, nor you damaging them by moving them far faster than their structural integrity will allow. Therefore, I would rule that time stop does affect the bomb—that is, it speeds the bomb up just as it speeds the caster up. At best, I would allow the caster to choose not to affect it, but then it would be impossible to move or otherwise affect, so the caster would be pinned in place by it and depending on how it was strapped to them, their ability to manipulate their limbs (e.g. for somatic components) might be restricted as well.
There is, however, another advantage of time stop here: with the right timing, if the caster puts the bomb down and moves away, or otherwise protects themselves (immunity to fire damage is trivial at levels where time stop can be cast; immunity to the concussive forces produced by a bomb is not so much trivial, but probably possible), and continues to affect the bomb (that is, it continues to move as fast as the caster), the bomb could explode while under the effects of time stop. While it’s a bit removed from the case described in the spell, where it is you who are trying to affect things, I would rule the bomb as limited by this rule:
While the time stop is in effect, other creatures are invulnerable to your attacks and spells; you cannot target such creatures with any attack or spell.
Therefore, the bomb could explode, but affect nothing (except perhaps the caster, though taking cover behind literally-immovable-and-invulnerable objects should probably work).
One thing about Time Stop that a lot of people forget is that you don't actually stop time with it. You just accelerate yourself to the point where everything else appears frozen. Any timekeeping devices would be unaffected unless you also rule that spells like Haste also accelerate them.