The Kirk speech is excellent. Although personally I'm torn between Picard's speech from Measure of a Man and the one from The Drumhead, but I think MOAM wins for me overall.
When you envision great Trek speeches, Beverly Crusher's face probably doesn't appear in your head... but maybe it should. "I am delighted that Worf is going to recover. You gambled, he won. Not all of your patients are so lucky. You scare me, Doctor. You risk your patient's lives and justify it in the name of research. Genuine research takes time. Sometimes a lifetime of painstaking, detailed work in order to get any results. Not for you. You take short cuts, right through living tissue. You put your research ahead of your patient's lives, and as far as I'm concerned that's a violation of our most sacred trust. I'm sure your work will be hailed as a stunning breakthrough. Enjoy your laurels, Doctor. I'm not sure I could."
The Hamlet “Q”uote from “Hide and Q:” “I see us one day becoming that…” I hope so Jean Luc… Still…one of my trek-like quotes comes from Justice League…as Clancy Brown’s Luthor delivers a similar justification to Amazo.
All time favorite, especially in the early seasons of TNG... Sometimes I look this up on YouTube because rewatching it makes me smile.... "Oh, I know Hamlet. And what he might say with irony, I say with conviction: "What a piece of work is man!...""
Not the greatest, but a very good one was Picard in The Drumhead... making his statement before his questioning to Adm. Satie
One of mine is from the last episode with Picard telling the pre Encounter at Farpoint crew that they basically need to take a big risk, but he trusts them with his life and they are sold on it.
I'm going a different route here & not go with the greatest preach, but what I consider the greatest speech, which I think is Spock's Eulogy from TWoK In that vein, I don't know if this other one qualifies, but I'll put it in here, because not only does it go largely unnoticed, but it is singlehandedly the 2nd hardest a Trek monologue has ever hit me in the feels, just after the above Guy had been in the service, living among humans nearly 2 decades before he met the 1st person who ever made him understand what a friend was
From the novelverse. Can't ID the speaker or the book but it was a real stirring speech about how the Federation treats its allies and even its enemies. Perhaps Bracco or another Federation president.