Can Slayers pick the rogue talent Combat Trick multiple times or not? I've seen long debates about this, but never found a good answer.
No, you cannot take Combat Trick multiple times.
The reason for this has nothing to do with anything in the Slayer class description; it has to do with the Rogue Talent class feature description.
A rogue cannot select an individual talent more than once. Source.
Some Rogue Talents, such as the Terrain Mastery talent, can be taken multiple times because they explicitly say so.
Special: A rogue can take this ability multiple times, each time applying it to a new terrain. Source.
Combat Trick does not contain any permission to take it multiple times, so it cannot be taken multiple times. That applies to anyone who can take Rogue Talents, not just Rogues themselves.
Whether Slayer talents can be used to take the same Rogue Talent multiple times if the Rogue Talent allows it depends on your interpretation of "it grants the slayer a new rogue talent," as discussed in Hey I Can Chan's answer. I would argue that each time a Rogue can pick a new Rogue Talent, even if he repeatedly takes Terrain Mastery, he is gaining new Rogue Talents each time. In my mind it works much like Weapon Focus: you take Terrain Mastery(Forest) as one Talent and later decide you want Terrain Mastery(Plains). They are in that sense separate talents. This is an interpretation, however, and there is no exact rules-as-written answer like there is for Combat Trick.
A slayer can pick the slayer talent rogue talent (combat trick) only once
The class feature slayer talent is introduced with the following text:
As a slayer gains experience, he learns a number of talents that aid him and confound his foes. Starting at 2nd level and every 2 levels thereafter, a slayer gains one [n.b. not new] slayer talent. Unless otherwise noted, a slayer cannot select an individual talent more than once.
Emphasis mine. Later, the slayer talent rogue talent says, in part, that
This talent can be selected multiple times; each time, it grants the slayer a new rogue talent.
Thus, while the individual slayer talent rogue talent can be picked more than once, each rogue talent picked must be different from one already possessed or else the newly picked rogue talent isn't a new rogue talent but an old rogue talent picked a second time.
(By the way, were each time, it grants the slayer a new rogue talent omitted, I'd totally agree with this answer, but allowing picking the same rogue talent multiple times ignores that sentence as, if it's read as permitting multiple picks of the same talent, the clause merely repeats the introduction to the class feature slayer talents, which already grants a slayer talent, in essence, saying A slayer gains a slayer talent twice for no reason. I know these folks are paid by the word, but that's why editors are salaried.)
In short, a slayer who picks the slayer talent rogue talent to pick the rogue talent combat trick is barred from picking combat trick a second time.
For comparison, with its full base attack bonus, two good saving throws, sneak attack, studied target, and track, the slayer is already a heady combination of ranger and rogue. Permitting the slayer also to take 10 bonus combat feats over the course of his career means virtually obviating the fighter, which Paizo has gone to great (some would say unreasonable) lengths to keep relevant despite Pathfinder's evolving character class landscape.
This talent can be selected multiple times; each time, it grants the slayer a new rogue talent
This is fairly unambiguous. Note that you gain a new rogue talent, not a 'new rogue talent not currently possessed', and especially note the lack of phrases like "do not stack'.
Compare Spell Focus's similar line:
You can gain this feat multiple times. Its effects do not stack. Each time you take the feat, it applies to a new school of magic.
When you can take something like this multiple times, the rules have to specifically call it out. When they want to limit how that multiple times works, they have to call that out, too. No such restriction is placed on the Rogue Talent Slayer Talent, and indeed it seems clear that some of the options (e.g. terrain mastery) are likely intended to be allowed to be taken multiple times-- no other class with access to favored terrains can't get more than one.
From a RAW standpoint, slayers are most indubitably permitted multiple combat tricks, allowing them to attempt to fill their more combat-focused role somewhat more handily.
Appended feat names are considered different individual feats throughout Pathfinder (this is a deductive argument). For example, the Investigator talent alchemist discovery (type-appended), and investigator talent rogue talent (type-appended) can be taken multiple times as long as its of a different type. "This talent can be selected multiple times; each time grants a new ..." alchemist or rogue discovery/talent. Ranger favored terrain [types] can also be selected multiple times as long as it isn't of the same type. The slayer's ninja trick [type-appended] "can be selected multiple times " The slayer's rogue trick [type-appended] "can be selected multiple times; each time, it grants the slayer a new rogue talent."
The rule "A rogue cannot select an individual talent more than once." is not abrogated by the talent terrain mastery [type], which can be taken multiple times as long as its a new terrain type. Likewise, the rogue ninja trick can be taken more than once without violating the individual talent rule.
The second argument is a language argument. We should expect Combat Trick (agile maneuvers) to be a different feat name, representing a different individual feat, than Combat Trick (alchemical strike) in much the same way as we should expect George Bush Jr. to be a different individual than George Bush Sr. When we append names to proper nouns, we are differentiating between those nouns.
Hence, I believe we can pick either a Slayer or Rogue 'combat trick (type)' multiple times as long as it is a different type each time. Likewise, we can pick the Rogue talent 'weapon focus (type)' or favored terrain (type) multiple times.
If however there is a circumstance where appended names represent the same thing, they do not stack (as is stated in the Ranger section of Terrain Master). If somehow there was 'Jungle-Forest', and we had both Terrain Mastery (Jungle) and Terrain Mastery (Forest), then we could use only one non-stacking Terrain Mastery (type).