We start looking at Yog-Sothoth. To defeat him you will need to gather 10 elder signs, and with a doom track of 10, you will need to make sure you up to the task. Should he awaken when you go to fight him, you must discard a trophy or be devoured (be removed from the game) so make sure you have some spare in case he wakes up. To defeat him you will need to roll 2 scrolls, and a 3 investigate, so he is no walkover. His innate ability is that if you fail an otherworld adventure the card is removed from the game, and if they are all gone he awakens, so he could wake up early as a nasty surprise. Keep an eye on that deck!
Yig has a doom track of 8, and needs 10 elder signs to defeat him. When he is the ancient one in the game, each time you lose on a cultist task or an investigator is reduced to 0 sanity you lose an elder sign, so a key hint is to keep an eye on your investigators, and if a cultist is on a challenge, make sure you go in tooled up to take him down. Should Yig Awaken, you will need 2 skulls to reduce his tracker and harm him. When he wakes up it can be deadly, as you must lose a spell, a common item and a unique item. If you do not you are devoured, so if you are just missing one of those things you will be killed.
When he attacks you must lose a clue token, or lose a sanity and a stamina, so he can really wear you down quickly.
Next up we have......
Azathoth..... he is the hardest in my opinion to take down, and I have not beaten him yet. He has a doom track of 12, and you need 14 elder signs to seal him in. If he awakens, the game ends, and the investigators lose as he devours the world. Best make sure you stop him first!
Ithaqua has a doom track of 9, and needs 11 Elder signs to be sealed away. He hurts you when you use items, which makes the game a lot harder, and when he awakens you may have to discard more items. When he attacks you have to discard a clue or lose 2 health which can quickly kill you. When fighting him you need a 3 investigate, a terror and a skull.
Shub-Niggurath gets my vote for coolest named Ancient one, the Black goat of the woods. A doom track of 10, whilst requiring 12 elder signs means you have to keep on top of it. His ability causes all monsters to have an extra terror dice needed to defeat them, making monster challenges more difficult. When he attacks each investigator must discard a clue or be devoured, and he needs 2 scrolls and a terror icon to take him on and reduce his doom.
Nyarlathotep adds the mask monster markers to the monster cup adding some more tough beasties to face. He has a doom track of 8, and needs 11 elder signs to defeat him. When he attacks you must discard a clue token or be devoured, certainly a challenge to defeat.
Rounding off the starter ancient ones we have Cthulu. The most well known of the Ancient ones, and as appropriate he is certainly not easy to beat. With a doom track of 12 and 13 Elder signs needed to defeat him he is not a walkover. His ability reduces each investigators maximum sanity and stamina by 1, which can be brutal to start as some of the investigators have very low figure in these meaning that failing those challenges is more punishing then normal. When he attacks each investigator must reduce his stamina or sanity by 1 or add 1 to the doom tracker. To battle him you must roll a terror, a skull and scroll. He is also my favourite artwork for the ancient ones.
Lastly we have Hastur, the king in yellow. His doom track is 11, and you must gain 13 Elder signs to defeat him. His ability forces you to add a doom token each time an investigator completes an otherworld task, so yes you may gain good rewards for them, but you bring his awakening even closer each time. When he attack each investigator must discard a clue token, or lose 2 sanity, and to battle him you need a terror, a scroll and X investigate, where X is the number of monsters in play, so make sure you keep an eye on those monsters and destroy them before he awakens, or he will be a beast to defeat.
The core game brings some excellent ancient ones to the game, all with their own unique challenges, and combined with different investigators can give you a huge amount of replay ability.