Well,for those who don't know what the big three is (Filthy casuals..) Allow me to explain.

Through the course of anime,for some reason there have always been about 3 shows that dominated the market at there time.Most people agree,that there were 3 cases of the big three.These 3 anime took over the world,at there time.Lets briefly go over them.

90's
Cowboy Bebop
Dragon Ball Z
Pokemon

These 3 shows dominated the market at there time.These were the big three anime you NEEDED to watch...Well,Pokemon annoyed the crap out of teens and adults,but lets move on.

Cowboy bebop was the underground hit,that helped impact the English dub market forever.Pokemon took over the children,and had them playing there games,and watching there shows forever.And Dragon Ball Z gave us teens the best fights EVER...At there time at least.Without a doubt,this is the most impactful of all the big threes.They changed the world of anime forever.

Early 2000's
Death Note
Code Geass
Fullmetal Alchemist 2003.

This is most likely the least well known of all the big threes.No doubt,they had quite the impact on the anime world. (I remember when the english dub of Death Note came out,and i was in highschool."ZOMG THIS IS THE BEST THING EVER"!)

However,here is the thing..They both have had there..Bad moments.And somehow,alot of people noticed them.Death Note starts to fall into a grave after "He who must not be named" dies.But the ending is very satisfying.Code Geass R2 Is..Its just weird."Lelouch has a brother.DERP!" "FAN SERVICE BABY! DERP" And the most subjective of them all,the 2003 FMA series has quite an odd ending.That is really bitter sweet.But,now we go on to the most well known of all the big threes.

Current Big Three (Not for long that is..)
One Piece
Naruto
Bleach

Every one knows these guys.They have made more money than all the other big threes before them,combined.They have completely taken over the market.To the point where they are at least heard of,by practically everyone.And they have created the phrase "Oh my god! A shonen that actually ends?!" They helped make the Shonen genre the most popular of all anime genres,which has made over a billion dollars.And once again,THEY WONT END! Oh you thought Naruto was ending?! HAH!

http://www.ign.com/articles/2014/11/07/naruto-manga-sequel-coming-in-2015

Also,the Bleach and One Piece manga shows no sign of ending.However,it appears they are finally starting to lose there steam,well..For Naruto and Bleach that is.You see right around when Bleach was rising,Fairy Tail was number 3,however,Bleach's popularity rose,and it eventually took over as number 3.Fairy Tail sort of lost its steam after that point,but is somehow making a recovery.In fact,Fairy Tail is actually rising again,on manga sales charts.Constantly staying in the top 50,even after Bleach stole its spot. (I personally still enjoy it more than One Piece.) After that,Naruto and Bleach were stuck for years,fighting to see who can take second place.

Meanwhile,One Piece was always mega popular in Japan.And due to the west,it got even more popular.Its the highest selling Manga of all time.In fact let me tell you something: Based on polls,and charts from Japan,it seems that 1/3 people in Japan,read one piece..That's insane.And it seems no matter how hard other anime try,they cannot take down these titans..Speaking of titans,lets get to the new big three.

But before that,where do these guys stand now? Well,the Bleach manga should be ending..Soon,i suppose.Naruto part 2 is ending..In fact,i think its already technically over.As Naruto and Sasuke already..I should shut up now,to avoid spoilers.Basically,every single chapter is already finished.The publishers just need to put them in a volume format.However,a miniseries/sequel is planned to come out in 2015.And One Piece is still moving with the force of a truck.And having a story predicted to be concluded,in the next ten million years,or so.

As i said,the current big three appears to be crumbling,slowly,but surely.It now seems that a new big three is rising.These new three are now.
Attack On Titan
Sword Art Online

Well..that's technically two,but anyway.

Yep,like it or not,these guys are starting to take over the world,just like the others.When Sword Art Online first came out,it was the biggest thing in the world.It was larger than life.It had such an epic feeling to it,but then..People found out it sucked after episode 14.And while it still remained popular,from endless drones still deeming it the best anime ever,no one had any idea what would come next.Wanna know what came next? Well,i think this picture somes up my answer.



QUE THE OPENING THEME!

Seid ihr das Essen? Nein, wir sind der Jäger!

TITAN BABY! TITAN! This anime is completely consuming the western,and eastern market.Holy gumballs,the amount of power this anime is gaining,so quickly is insane.And guess what,in just one year,it is now in the top 20 anime series in the world,in terms of sales.Already making millions of  dollars,on the anime,manga and other merchandise.WHAT?! Thats right,in the course of just under a year,this thing took over the market.And its taken over the manga world to! When the 6th episode of the anime came out,the amount of manga volumes sold went up by 88%!

And this begs the question,why did these two shows get so popular,in the first place! Well that's something i can make in another discussion.Right now,we are dealing with the question at hand,and that is this: What is the next anime,that completes the trinity? What is going to be the next big thing,that completes the set.What will be the next big three? What will be the big thing everybody will be talking about?

Well,i don't have an answer for that,so i will leave it to you guys to give me an answer.See yah next time. *Waves*

Hm..You know what,i think i have a list i can make."Top 5 reasons why Attack On Titan is so popular." Yeah,that sounds about right.

I've only ever heard of One Piece/Bleach/Naruto being referred to as The Big Three/Holy Trinity. I suppose you could say that every generation had some anime that were well known and dominated television more than another. Your criteria for listing these baffle me however, for example with Cowboy Bebop. I know many people consider it to be a great anime and a work of art, but I don't think that is what dominating the market really is. Though to be fair I was a bit too young to have noticed if it was that popular at the time. The three certainly made a big impact in the industry on anime but impacting and being well known are not always one in the same.

Your early 2000's list is where I really am confused at what this criteria is. Death Note was popular for sure, and Fullmetal Alchemist (2003) and Code Geass were also popular and almost universally considered great, but they didn't make a huge impact in the industry. Anime that do exceptionally good will always be examples to follow in making a good one for the future, but that doesn't always mean what they were good for was groundbreaking.  Not to mention One Piece started in 1999, it was known well before those three you listed. At the time they were released they were probably talked about more than One Piece but that's because One Piece was not new and still running regularly. Short anime tend to have more hype and flare around them, but die out faster as they end (people will still talk about them but not on the same level). While in the case of the Big Three they are all long running and continue to flare if only of a normal level.

Then you say that the Big Three are the reason shonen are popular, and I disagree. Dragon Ball Z in my opinion was what made shonen what it is today. Even the creator of One Piece, the longest running of the three said his inspiration for it came from Dragon Ball Z, so it's safe to assume whatever popularity One Piece draws for shonen is directly associated with Dragon Ball Z.

I cannot speak for Naruto or Bleach because I have not watched them fully (only watched the original Naruto and around 200 shy episodes of Bleach), but at the time both of them needed filler to buffer out the chapters that were coming out. I watch One Piece myself though it is starting to slow a little as the manga chapters are closer to catching up with the anime. As far as story goes? There's still a ton of things left to be explained. If a problem for Naruto or Bleach is the lack of plot left to explore, One Piece has no trouble with that. I know both are nearing an end (unless Naruto is actually getting a 3rd series, in which case I guess not) but even then Fairy tail while popular has yet to really take that spot as one of the leading three, though I see it as a strong 4th in terms of popularity.

Now with your current "Big Three" I am again seeing some issues with whatever criteria you have for them if any. Sword Art Online was indeed popular like your early 2000's listing, but again it makes little to nothing groundbreaking in the industry itself. I really believe it's only popular due to a lot of western anime viewers who also play MMOs so it's something relate-able. It had it's flare, but I don't see it talked about much at all anymore. Not only that but it hasn't even lasted long compared to the current shonen around, though I think it has enough source material to continue making episodes, to the point where it might reach that point (I only know that it had a novel so it's just an assumption for that).

Attack on Titan is a different case from before, because it is immensely popular in the west compared for the short time it has been around as well. It has that popularity and impact that Cowboy Bebop had (though comparing them would be debated somewhere else), however it also shares the same issue of not being a long series. It could probably churn out another season but in terms of length, but it couldn't run with what the current series are doing.

One Piece, Bleach, Naurto, are all very long series and popular to a point where most series today will just not reach that point. Fairy Tail would be a contender for it if one were to end, but there isn't another popular long running series still around that is still relavent today. Hunter X Hunter is probably the closest but as it has recently ended as well due to lack of source material. I cannot think of another anime currently beside Fairy Tail that has both the popularity and substance to sustain itself as a "Big Three". In the end I still really cannot figure what your judging an anime on to be considered, as they all have different reasons while the current three ones all share something similar, being a long running shonen.

Wish DBZ was still top 3...
Bleach...ah Bleach and Fairy Tale.
They seem to remind me of each other, for obvious reasons, but Bleach just never...appealed to me. I can't stop watching now because I watched an AMV(oh god the spoilers) and saw some cool stuff, and I don't want to skip to it. I'm on EP 86 right now. Please Jesus, take the wheel.
As for Fairy Tale, its a lot better, Erza's great, and Gray's awesome. The stuff in it is flippin' amazing!
Attack on Titan and SAO though...
SAO, in my opinion, is the second best anime I have ever watched. It's only problem was shortness and that's only because my freind said they were there for (SPOILER ALERT) 3 years. So I though, "yay, 1,000 episodes!" And I got 40 or so.
Attack on Titan is beautifully made, with great characters and stuff to, and epic plot twists that everyone was expecting!
But I have to say that Ichigo, Erin, Rin(Blie Excorsist), Natsuu, etc, are just to mainstream. They all have nearly the exact same personality.
Loyal to freinds
Determined, despite being an underdog
Get stronger insanely fast(Natsuu being an exception)
Prone to rage fits
Corny quotes
Yep. Somes it up.

DBZ though, Goku, Vegeta, Gohan, Goten, Trunks, all extremely unique characters with a great plot and epic battles, takes the top for me.