This can be from any game of the Fallout franchise.

In first place is the CZ57 Avenger A.K.A "that big ass minigun that murders anyone dumb enough to run towards whoevers using it.".
A close 2nd would have to be Oh Baby!  Because It can obliterate pretty much everything and everyone.
And in 3rd place is Euclid's C-Finder cause there's nothing like calling down death from the heavens themselves.
(You don't have to do a top 3 like this I just felt like sharing a bit more than one.)

This Machine. Since its a M1 Garand....and i loooooove the plinking sound during reload

Anti-Materiel Rifle since it pretty dismembers anything that moves.

FO3:

Dart Gun
Railway Rifle
Gauss Rifle (< This being the best)

FNV:

Anti-Materiel Rifle
Anti-Materiel Rifle
Anti-Materiel Rifle

That gun. for all your bladerunner needs then they had to ruin it and add a common version in GRA

Well... From Fallout New Vegas It whould be....

That Gun - because it's amzing futuristic gun and Blade Runner reference. It gives you the punch of revolver with amaing design. i specially love it with Toasty replacer for it <3

Gobi Scout Rifle - It is shorter and more dangerous version of sniper rifle and If We are talking about sniper rifle I rather have small compact one (weeeell... Anti Material rifles are another love of mine >:] ) Btw It have very nice camo on it.

In Fallout 3 It whould be...

Xua-Long Assault Rifle - It is very powerful gun Named after chinese dragon... And It is completely amazing. Tough variety of unique guns in Fallout 3 makes me cringe ;-; I was thinking on making a mod that whould retexture or replace it with something more standing out :/

I've always liked the Mysterious Magnum.

The Medicine stick is a great weapon simply because of how freaking powerful it is. Great accuracy and damage combined with a great iron sight makes it a perfect medium range weapon.

Ranger Sequoia is probably my favourite weapon when I can get it. Big, powerful, badass looking, everything a beastly revolver should be. Combine it with swc rounds and you've got yourself a deathclaw killer

The Medicine Stick. The overall damage and DPS is really outstanding compared to its normal variation, the "Brush Gun" and it looks really cool with that little tribe thing on its Stock

My favorite weapon would have to be Mercy because nothing says F*** You like a 40mm Grenade MG.

Mine is YCS/186 with max xharge ammunition. Nothing kills Deathclaws like 234 base damage with a sneak attack bonus. Second place would be the light machine gun because lets face it, sending hundreds of 5.56 match rounds downrange to even heavily armored foes is always satisfying. Third would be the hunting rifle, assuming all three mods are attached and using JSP ammo.

My unique weapon of choice is the Terrible Shotgun. It is the one I use the most. Powerful and comes in handy in a mob fight close quarters.

Favorite Unique Weapons:

F3:

Lincoln Repeater.

I freaking love that rifle shame I got it early on in F3 so that it wasn't really useful at end game.
Dunno why exactly I love it some much, because I'm a Abe Lincoln fan or just because it's an awesome rifle. Really can't say.

Jingwei's shocksword:

Does a lot of damage and the whole effects are so freaking awesome.
Eventhough I prefer fire over lightning, this weapon was my favorite.
I've also noticed it often acts like an EMP, some robots shut down when I killed other enemies with it.

FNV:

That Gun: Awesome pistol that really packs a punch, even more usefull when using a mod that improves it :p.

Less unique:

Anti Materiel Rifle:

Freaking badass rifle, what more is there to say about it? I do love the modded version Ranger AMR, it's stronger and it looks beautiful with its details, especially the inscription: "Against all Tyrants"

Ranger Sequoia:

I freaking love that revolver, it's just plain badass and I think it fits with the whole ranger outfit.

Fallout NV: This Machine because of it's look and has pretty decent damage.

Kinda(?) unique weapons: Ranger Sequoia, has high damage and nice look to it.

I've always really liked Dinnerbell simply as it is a slightly cooler looking retex of the hunting shotgun!

Fallout 3:
Lincoln's Repeater, an obvious choice.
A3-21 Plasma Rifle. This thing shreds through everything, even with a low energy weapons skill

New Vegas:
All American. I like having the scope and it's pretty powerful
Dinner Bell. Can never go wrong with an awesome looking shotgun

The holy hand grenades of course!

F3: Lincolns Repeater
New Vegas: Medicine Stick

My favorite from fallout new vegas is the broad machete

Post 2:

Whoops didn't mean to double post

EDIT:

dont know if your trying to be a troll or replaced the text in your original double post with "Whoops didn't mean to double post" be careful i gave you alot of warnings in the past for double posting the next time you'll be given a minus rep as it will work as a constant reminder.

:no minus rep given:

FoVet20

FO3:
Alien Blaster
Alien Blaster
Alien Blaster

FO:NV
Alien Blaster
Alien Blaster
Alien Blaster