Updated: Oct 27, 2018
Itchlarin's Little Helper actually has no skill requirements whatsoever, and it can get you from level 1 all the way to level 19 Agility, 20 Thieving, and 19 Woodcutting. So, those are three kind of slow skills at first, so it is a pretty decent boost.
Murder Mystery (Crafting) Now, there are not many Quests for Crafting. The main one here is Murder Mystery, and this one gives you 1406 XP in Crafting, which really isn't much. At low levels you do get pretty good Crafting XP if you cut Opals. The reason why I'm putting this in is because it's such an easy Quest that doesn't take a lot of time to complete at all.
Mourning's End Part 2 (Agility) Now, the Mourning's End Part 2 Quest actually gives you 20 000 XP in Agility, which is about half an hour of Agility right there. But, the Quest itself is very long. It's very tedious, and a very annoying Quest. But, the 20 000 Agility XP kind of makes up for it a little bit.
Fairytale Part 1 (Farming) Fairytale Part 1 has no Farming requirements at all, and at the end, it gives you 3500 Farming XP. Let's say you were at level 1 at the start, this Quest would get you all the way to level 17 Farming. Forgettable Tale of a Drunken Dwarf (Farming) Then with level 17 Farming, you can move on to the Forgettable Tale of a Drunken Dwarf, which actually requires level 17 Farming. That gives you another 5000 XP in Farming, getting you straight to level 25 Farming. Farming at the low levels, as most of you would know, is really, really tedious and slow. So, getting to level 25 straight away gets you to doing some tree patches and that speeds it up a lot.
The Sea Slug (Fishing) The Sea Slug Quest gives you 7175 Fishing XP, and it actually has no Fishing requirements at all, and you only need 30 Firemaking to do it, which is very easy to get, and you'll get from level 1 all the way to level 24 Fishing with that XP reward. It's a super short Quest.
Eadgar's Ruse Quest (Herblore) The Eadgar's Ruse Quest requires level 31 Herblore, and you actually get 11 000 Herblore XP for completing it. So, that will boost you from level 31 all the way to level 36.
The Digsite Quest (Mining) Now, in terms of Mining, the Digsite Quest gives you 15 300 XP in Mining, and it has no Mining requirements at all. That would get you form level 1 all the way to level 31 Mining, which is awesome. It's actually a fairly easy Quest as well. It does have a couple of other requirements, but no Mining requirements.
Holy Grail Quest (Prayer) Now, Prayer is a skill that is very, very quick for a normal account, but iron men can use these Quests because they're very handy for boosting your Prayer up at a low level without the Ectofuntus, or a Gilded Altar on an iron man. So, the whole Holy Grail Quest gives you 11 000 Prayer XP, and you don't need any Prayer level requirements for this. That would get you to level 28, all the way from level 1 Prayer. There are some other Quests that give Prayer XP, but they don't really give that much, so I'm not going to include them.
Eyes of Glourphrie (Runecrafting) Now, Runecrafting is really, really slow at low levels. So, the Eyes of Glourphrie Quest has no Runecrafting requirement, and it gives you 6000 Runecrafting XP. That would get you from level 1 all the way to level 22 Prayer. What Lies Below (Runecrafting) The What Lies Below Quest requires level 35 Runecrafting, and it would actually get you a couple more levels since it give you 8000 Runecrafting XP. So, that is pretty good.
Wanted (Slayer) Now, for Slayer XP, there aren't many Quests that give a lot, apart from the Monkey Madness 2 Quest, but the Wanted Quest gives you 5000 Slayer XP, with no requirements at all. So, if you were level 1 when you do the quest, you'd get all the way to level 20.
The Knight's Sword (Smithing) Smithing is another one that has a pretty good Quest in it, and that is The Knight's Sword Quest. This quest gets you 12 725 XP, and that gets you all the way to level 29 Smithing with no requirements. So, that is very, very good for new accounts, and it's a very quick and easy quest.
Contact (Thieving) Lastly, there is Thieving. The Contact Quest gives you 7000 Thieving XP with no requirements, and that will get you to level 23 Thieving from level 1. The Feud (Thieving) Then there is the Feud Quest, which gives you 15 000 XP, but it does require you to have level 30 Thieving in order to do the quest. But, 15 000 XP is a lot of Thieving XP.
Now, in the final part of this article, I'm going to talk about quests that give you XP lamps. The Tourist Trap First up is the Tourist trap Quest, and it can give you 4650 XP twice. So, you have two options to get that much XP, and the skills you can pick are Agility, Fletching, Thieving and Smithing. There's actually no Agility or Thieving requirement for this quest, so putting it on Agility would be really handy. Recipe for Disaster Quest The next one is the Recipe for Disaster Quest. It's a very long quest with a lot of high requirements. But, at the end, you get a 20 000 XP reward for any skill that is above level 50. So, 20 000 XP is pretty good for something over level 50.
The Shadow of the Storm The Shadow of the Storm quest has very little requirements, and it actually gives you 10 000 XP in any of the combat stats. It's up to you which one you want to decide on, there's no level requirement behind it. So, you can really boost your levels with that. One Small Favour The One Small Favour Quest gives you 10 000 XP in two skills, and you can actually pick any skill you want that is above level 30. So, that is another really, really good one for leveling up your lower skills. The Legend's Quest The final one is the Legend's Quest, and this one gives you 7650 XP in four skills. So, you get that much XP in either Attack, Strength, Defence, Magic, Hitpoints, or Prayer. So, all the combat skills. And then you actually have a choice of other skills such as Thieving, Agility, Smithing, Herblore, Crafting, and Woodcutting. I would personally put them all on Agility, because Agility is really hard to work on.