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The galaxy is a big place. But the universe is much bigger. Still, most of Star Trek canon seems restricted to exploring our own galaxy - and even then, only a small part. Sure, there's limitations to exactly how fast Federation ships can travel. But there have also on occasion been times where external forces or influences have thrown ships far beyond their normal range, forcing them to travel much faster than anyone imagined possible.

This question addresses visitors coming from outside our galaxy. However, I'm interested in any incidents where Federation crews or other Milky Way natives have been flung beyond its reaches. Also, my interest is particularly in normal space travel - not places like fluidic space, alternate dimensions, etc.

Excluding travel to other dimensions, universes, or timelines, or fluidic space, has there ever been a Star Trek episode (in any series) that included travel to places beyond our galaxy?

Yes, they have.

• In VOY Threshold Tom Paris and Captain Janeway go warp 10, which supposedly allows them simultaneously occupy all points in the universe at once.


• In TNG Where No One Has Gone Before the Enterprise travels to a distant galaxy, with the assistance of The Traveler.

Extra-galactic species on Memory Alpha has a list, but presumably some elements in the list don't fit your limitation of "no other dimensions".

I added other elements from other sources as well.

Questionable:

• Species 8472 ("fluidic space")


• Q (extradimensional plane of existence )


• Space amoeba (unknown origin)


• Sphere Builders ( transdimensional beings )


• 
  Talosians (disputed whether Talos IV in Vernal Galaxy was actually part of Milky Way or a separate galaxy)


• Onaya claimed she could spot a creative soul "a galaxy away". If taken literally, this might indicated that she had at least traveled outside the galaxy, possibly having originated there.


• 
  Nagilum (formless immortal being that lived outside of the universe)

Fully qualified:

• Douwd




• Kelvans ( planet Kelva, located in the Andromeda Galaxy)




• Nacene




• Old Ones




• Ornithoid life forms




• The androids of Planet Mudd were said to be from the Andromeda Galaxy. (TOS: "I, Mudd").




• Spock also speculated that the Neural parasites encountered on Deneva might have originated in another galaxy.




• Enterprise-D traveled to Triangulum Galaxy (sent by The Traveler, in TNG: "Where No One Has Gone Before")

Non-canonical (from books, via Memory Beta)

• Hive Species




• Nuyyad ("The Valiant" book)




• Several species met and travels to Andromeda Galaxy


-

In TOS "By Any Other Name", the Enterprise does leave the Milky Way on a voyage to the Andromeda galaxy, but they turn around before actually getting anywhere. Yes, the Kelvans "souped up" the Enterprise's engines to go much faster than it was otherwise capable, and thereby shorten the expected duration of the trip, but it was still going to be multi-generational.

So, strictly speaking, Federation citizens (Earthlings + Spock) have left their home galaxy, but no, they didn't actually arrive at any extra-galactic destination, or even waypoint, in the course of the journey.

The confirmed trips outside the Milky Way as far as I can tell:

• TOS S02E21: "By Any Other Name": The Enterprise breaks through the Galactic Barrier and fly in direction of the Andromeda Galaxy after being commandeered by natives from Andromeda.




• TNG S01E06: "Where No One Has Gone Before": The Enterprise is taken to the M33 Galaxy then to what is described as the edge of the known universe by the Traveler.

The Voyager episode "Treshold" (S02E15) is a tentative one: theoretically, anything reaching Warp 10, which is an infinite velocity thus unattainable except by the likes of Q who have little regards for our laws of physics, would find itself at every point of the universe at once.

While the episode states that the shuttle Cochrane did reach Warp 10, nothing is said about going outside the galaxy. Wile Tom Paris states that he could see other galaxies, it's never said if he could see them from the inside or from the outside, as if looking from the Milky Way towards them. And the data they retrieved from the shuttle's computer contained detailed information about the sector the Voyager was currently in, but nothing is said about data from places outside the galaxy.

I know a couple times in the series some have been but very shortly and then placed right back. I know as we see it that Q is possibly the most developed species of the entity of endless space and if they are not then the whole concept of something even more powerful as Q is unconscionable. A being that can go anywhere, do anything, can change anything, can even bring someone back from death (a state of nothing), and for Q to not even know it is just wow.

Now that to say this. When Q was made human his IQ was still like 2600 and that was in human form, then even told picard that humans may even surpass the Q. and lastly I cant remember what episode in TNG but Q said you could see everything in this universe and can travel simultaneously (reach the end as the moment you start), and yet you still couldnt comprehend what is out there.

If Q has lived for eons of eons and is just now starting to bored then there must be one hell of an universe. Then you have to take in effect if there is trans universe? meaning like how 4 quadrants make up our galaxy then what if their is there is 4-8 universe to divide countless other universes?