Special Monsters (only 1x scoutable)
The creaters of this side:
Note: Here I want to list any special monster, which can be obtained only once.
Before beating the game
After beating the game
You'll need to visit places, which are rainy. My order was: Coast, Plain, Jungle. Then, Captain Crown will appear.
You can fight the 3 first battles after beating the B arena/unlocking Cliff aera
1st Battle: Orc (91 hp) x2 & Seadog (104 hp) (Reward: Exp. 250, Gold 424, can drop Iron Lance or Copper Sword)
2nd Battle: Seadog x2 (104 hp) & Cross Bones (292 hp) (Reward: Exp. 343, Gold 920, can drop Copper Sword or Rapier)
3rd Battle: Cross Bones x2 (292 hp) & Captain Crow (1003 hp) (Reward: Exp. 1977, Gold 1480, can drop Rapier)
After beating the game, go to the cave and look at the wooden panels
and talk to the Bopping Badger* to unlock the 4th and 5th against Crow.
4th Battle (at Plains, rainy): Bilhaw x1 (1773 hp) (Reward: Exp. 5316, Gold 1020, Steel Claws)
5th Battle (at Cliffs, rainy): Captain Crow (3007 hp) (Reward: Exp. 9793, Gold 520, monster Captain Crow)
After the battle, Crow joins you.
*This is the Bopping Badger you need to talk to.
EDIT: Talk to him again after defeating Cpt. Crown for a Gold Card !!!
At the final area in the new world, go east of the entry of the Palace and take the teleporter
Lleviathan can be fought at the coast. It will join you after the fight.
After clearing the new world, talk to it in the elder house
and he will join you.
After clearing the new world, Wulfspade Ace will go back to his own world, once he returns you'll be able to find him in random spawn points on each map by seeing a huge metal type mirror. Once you open the mirror you'll be teleported into a battle with a different Incarni form for each battle. Once you find and beat him 4 times he will spawn in the ship area to the left side of the map up the slope by a treasure chest. Talk to him when you spot him(he'll be in his lil pup form) and then he will join you. Also, note he can be used for the guest question for Marsha(the mud mannequin)'s quiz.
For Incarni Location : 
After clearing the New World, go to the cliff, and Warp to the summit and go to the cave.
Beat it in 10 turn or less (harder than it sounds!).
After clearing the New World, talk to the Sanguini at the left of the throne and answer yes
You will have to fight 100 battles against random opponents. You can't save or heal between the battles.
You obtain him after losing or abandoning if you have won at least 100 battles.
You also obtain at least 250 000 GPs
You must NOT have beaten him before doing the 100 battle or you won't obtain him.
Feel free to add how to get it, if you know. =)
EDIT: Stormsgate Citadel is only attainable through wi-fi. You have to win the WMC.
Watch here: http://dqmj2p.wikkii.com/wiki/Only_Rank_S_%26_Rank_SS_Monsters_-_Combinations#Rhapthorne_.28form_1.29
After clearing the game (!), buy an egg for 110.000 G in the shop in "New World" and bring it to New World Part 6 (added by WPL). Walk up to the shrine, check the nest and answer "yes." The game will save. According to the green pipit, you have to do a certain number of battles to get the egg to hatch. I don't know what the number is, but it's not a few. Do some synthesizing and/or training for a while and check back every so often and it should hatch.
Watch here: http://dqmj2p.wikkii.com/wiki/Only_Rank_S_%26_Rank_SS_Monsters_-_Combinations
Feel free to add how to get it, if you know. =)
You can scout one Titanis at the very top of the Cliffs after beating the game. It's easy to synthesize, if you'd rather do that - you can make Titanis by combining either an Atlas or a Trywinder (big orange snake on the Coast) with a Moosifer, and any of these monsters can be scouted at this point in the game. Titanis is needed to synthesize Estark.
Bosses (MORE THEN 1x scoutable)
Note: There is only one other rule that might make people think it is a 1 time only thing.
You can only have 1 of each the bosses in your 'inventory' at any given time.
E.g. if you already have bjorn in your monster park or party, he wont spawn again.
But if you breed him with something else, he will be there again.
(by SOuimet): Technically, it's impossible to have a "Best Wiggly" or "Best Bjorn",
as it's impossible to breed one or to scout more than one at a time.
However, in the library, there are the rank S "Best Wiggly", "Best Bjorn", etc.
Some, like Khalimari are breedable though, so it could be possible to make a rank S "Best Khalimari",
with only some improved stats / limits and no new traits.
Empyrea = Daytime, no strong weather. Just examine the eggs at the top.
Missing Lynx = Night time, not raining. Just approach him in his cave, fight his tail then you can scout in the next fight
Bjorn = daytime, no strong weather. either use the campfire when it is lit 1 zone from the warp platform, or find him wandering around the other zones
At Coast (Part 8 - Cave), no requirements, you dont even have to beat all the tentacles, just warp to the platform and kick his face
Only at night in the cave, walk in his mouth to start the fight. You CANNOT scout the one that slides around the jungle without cheating.