Well, I'll attempt to list as many ways to get home as I possibly can. It's been awhile since I've seen Star Trek: Voyager since I don't own it on DVD yet and it only airs in the wee early hours of the morning and I value my sleep.
So I'll attempt to get as many as I can. Anyone can feel free to add to this list or adjust any facts I don't get exactly correct.
Season I:
"Caretaker": The Caretaker Array itself, which brought Voyager to the Delta Quadrant. Janeway ordered the array destroyed to prevent the Kazon from using it against the Ocampa.
"Eye of the Needle": Voyager encounters a micro-wormhole that leads to the Alpha Quadrant. Only problem is, it leads to some twenty-odd years in the past. The crew delivers messages through a Romulan scientist, but their messages are never delivered due to his dying before Voyager was scheduled to leave on its mission.
"Prime Factors": An alien race with technology that could cut Voyager's travel back to the Alpha Quadrant in half refuses to give her their technology, a request that Janeway honors. However, when B'Elanna, Seska, and Carey steal the technology, it was found to be incompatible with Voyager anyway.
Season II:
"The 37's": Voyager encounters a group of humans who were abducted from the Alpha Quadrant. But the race who deposited them there has apparently been killed off (I think this is how it went).
"Cold Fire": Voyager encounters Suspiria, the Caretaker's mate, who seems more interested in destroying Voyager than in helping them return home.
"Threshold": Tom Paris breaks the warp 10 threshold in the aptly named Shuttlecraft Cochrane, allowing him to occupy all points in the galaxy at once. However, it turns out that achieving warp 10 has some bizarre side-effects and is deemed too unsafe to take the ship through the threshold.
"Death Wish": Voyager finally encounters Q, who of course possesses the ability to send Voyager home. However, after they rule in favor of Quinn to achieve humanity, Q leaves them where they are. I think Q even offered some rationale as to why he wouldn't actually send them home.
Season III:
"False Profits": Voyager encounters Drs. Arridor and Kol, two Ferengi from the Alpha Quadrant who were stranded on a planet after the unstable Barzan wormhole shifted its location. Voyager discovered the event horizon was back and managed to get Arridor and Kol back to the Alpha Quadrant but the event horizon shifts again before Voyager can go through. The ultimate fates of the two Ferengi doctors remains unknown.
"Future's End": A two-part episode where Voyager encounters Captain Braxton and the timeship Aeon. They are sent back in time to Earth's past. However, after a devastating event is avoided and they meet a Captain Braxton who has no memories of the event, he informs them that the Temporal Prime Directive prevents him from depositing the ship anywhere other than the point in the Delta Quadrant at which they left.
"The Q and the Grey": Same as "Death Wish." Q could but doesn't send them home.
Season IV:
"Hope and Fear": The crew encounter Arturis, who leads them to the U.S.S. Dauntless which can supposedly take them home, using quantum slipstream technology. However, the ship turns out to be a trap and quantum slipstream drive is too unstable to take them very far (though they do shave a bit of time off their journey).
Season V:
"Timeless": The dangers of a quantum slipstream drive are realized when Voyager crash lands on an ice planet, killing everyone aboard (Chakotay and Kim are the only survivors, having safely made it in the Delta Flyer). Chakotay and Kim send a message back through time urging Voyager not to use it.
"Dark Frontier": Janeway attempts to steal a transwarp coil from a Borg Cube to shorten their journey home, but soon realize the Borg Queen was luring them into a trap to force Seven of Nine to rejoin the Collective. They gain a little ground through the transwarp conduit but not as much as they'd hoped, due to a battle with a Borg ship.
"Relativity": Same deal as in "Future's End". They encounter Braxton in a new timeship, the Relativity, but the Temporal Prime Directive prevents him from sending Voyager home.
Season VI:
"Equinox": Having encountered the Equinox, a Federation science vessel brought to the Delta Quadrant by the Caretaker, they discover that the Equinox crew has been killing lifeforms native to the area in an attempt to return home. Needless to say this does not sit well with Janeway and she refuses to participate, going so far as to stop the Equinox from doing the same.
Season VII:
"Inside Man": A hologram of Barclay arrives with a dangerous plan to get them home, creating a geodesic fold between a red giant in the Alpha Quadrant and in the Delta Quadrant. However, the real Barclay was unable to send his hologram; this is a trick by the Ferengi to steal valuable Borg nanoprobes from Seven of Nine.
"Q2": Same as in "Death Wish" and "The Q and the Grey," another encounter with Q.
So there we have it: 17 distinct opportunities to return home, and not all of them due to refusal to accept the help. There are a number of reasons why they didn't accept the technology, or why something didn't work out, etc.
Honorable mentions:
"Endgame": I didn't mention this one because it's the final episode, and most of us know how it turns out. But I didn't want to spoil anything for anyone who hasn't yet seen it.
"The Gift": Kes, having evolved into a higher form of life manages to send the Voyager some 9,500 light-years closer to home, knocking about ten years off their journey. But this list was more about failed attempts to return home and about not being able to have the technology. So I didn't include this one on the list.
I hope there are some out there who find this list interesting and insightful, and again if I missed anything please feel free to let me know.