Yes, but it's more work than you'd think
You could keep levelling, but the game starts breaking down. You start running out of moves that you can take and you rarely ever fail rolls because your stats are all in the positive. The engine runs out of steam and the game starts to be boring.
So you can, but you would have to start houseruling lots of bits of the game to make it work. For a lack of moves, you could either add more multiclassing options at level-up, or you could go big and add a whole new tier of level 11+ "epic" moves for each class. (You can write new moves one at a time as each new level is gained though, so you don't have to write a whole tier all at once.)
For bonuses, I have no idea how to fix that: making things harder mathematically doesn't actually work very well in Dungeon World, and otherwise there's no effective way to make the 10+/7–9/6− continue to work and trigger GM moves often enough keep the game rolling. Remember, the GM doesn't even get a turn to say what happens next unless the players give it to her, either by looking to the GM for what happens, or by rolling a miss. If the players never miss, that's half the GM's input into the game gone, and the campaign's momentum with it.
So yeah, you can, but at some point you're playing a different game design-wise, and you're in unknown design territory, piecemeal crafting a brand-new game without any useful guidance from the text of Dungeon World. Go for it if you like! But be aware of what you're getting yourself into before you go for it.
The existing options are better than they seem at a glance
First, make sure you're not accidentally robbing yourself of 10th level entirely. You said
… when you hit level 10 the manual says you must…
But that's off by a whole level: you only follow the Beyond 10th Level rule when you are already level 10 and you would advance to 11th level. I've seen numerous posts scattered around Dungeon World discussion sites where people make the mistake of following this rule as soon as they hit 10th level, which does make a character's career significantly shorter than it should be.
With that out of the way, it's worth taking another look at the existing 10+ level-up options. Like most parts of Dungeon World, they contain way more than they appear to at first glance. I don't know what your particular motivation is for wondering about continuing past level 10, but as this is a pretty common question about Dungeon World, I'm going to cover all the bases and let the reader make the final evaluation with more information.
Retire to safety
This one is pretty straightforward: you retire the character and make a new one. But note that the GM is still beholden to the GM's Principles when you make this move, in particular Be a fan of the characters and Think offscreen, too. Your retired character doesn't suddenly stop existing – they don't suddenly leave the game. They do, in the fiction, literally what the move says: the person retires from adventuring to a place where they are (relatively) safe from the dangers of the GM's moves. They are still in the world, and the GM is still required to be a fan of them and to think about them while they're offscreen.
As well, when you make this move, you still have to follow the rules: the move returns to the fiction. That means that when you trigger the move and choose retire, you have to, nay, you get to say what that looks like. Do you get given a title for service to the realm? Do you become headmaster of a school of wizardry? Do you found a new branch of the church in a new land? Do you settle down into a new dwarf kingdom under the mountains? What awesome thing does retirement look like for your character? You get to work with the GM to figure that out.
So what retiring means is your character becomes an awesome NPC in the setting. Retiring a PC to safety is your way to make one last big change to the setting. That's pretty sweet, and not an option to be overlooked.
Take on an apprentice
You get a second character, your first stops advancing, and your new one gets to restart the whole rags-to-riches story again while your original PC keeps on being their awesome self. This may seem like a step backwards, but it's not: it's getting to have extra spotlight and more opportunities and ways to control the fiction.
If starting over as a 1st-level scrub seems like no fun, remember that Dungeon World is designed so that you can play with a mix of levels and it works just fine. It's even a rule to emphasise the point: when a new character comes into the party after your last character dies, you start at 1st level no matter what level your dead character was. I know lots of people doubt that would work, but trust that the game knows what it's talking about and that it does work. This ain't no D&D 3.x, where level disparity can be a huge problem.
So, consider the benefits:
You get to start with a 1st-level character over again. You get all the things a 1st-level character does when they enter an existing group: new questions from the GM that you get to answer and shape the world with, faster leveling because you're going to hit those 6− rolls and mark XP more than your fellow PCs will, new-to-you awesome moves that get to influence the direction of the game in ways that you haven't been able to with your last PC.
You get all that, and you get to keep your fiction-making juggernaut of a 10th-level PC. You can team up on an enemy all by yourself now, you get double the influence over the direction the game goes next, and you can be in two places at once. That is super-powerful, much more than the simple words "Take on an apprentice" might seem to be at first glance.
Change entirely to a new class
This has all the benefits of a brand-new PC, except that you keep a tonne of stuff from your previous class. Consider it the super-multiclassing move.
What do you keep?
- ability scores
- race
- HP
- and whatever moves you and the GM agree are core to who your character is
That last bit is huge. Remember, the GM is rules-required to Be a fan of the characters. Being stingy about agreeing what is core to who your character is, isn't any part of the GM's Agenda and Principles, and a GM who's being a hardass about what you get to keep is not following the GM's rules.
You get to keep everything that makes you who you are. Are you a master thief, with your own thieves' guild, an influence-peddling racket, an infamous reputation for impossible burglaries, and basically own the kingdom with a web debts and obligations owed you? You keep all that, plus the moves that suit that identity, even if you suddenly develop an interest in enhancing your legend with the ability to turn invisible and use other spells useful to a master thief. That might mean you get to be a wizard plus the Thief moves Flexible Morals, Poisoner, Wealth and Taste, Connections, Escape Route, and Heist (assuming you took a bunch of moves with those among them, and those are the ones that match where you've gotten in life). You lose everything that isn't essential to who you've become, so maybe you lose things like Evasion and Serious Underdog because you haven't been known as a scrappy brawler since your early days.
Like I said, change class is the super-multiclass move. You get to keep a lot, a lot of stuff from your previous class, and then you get the fun of playing with a new class's toys all over again, while still keeping your best toys.
In summary
Yeah, you can keep levelling past 10, but the math and other intangibles of the game are designed around the 10th level cap. To best keep going past level 10, you can probably just keep playing while keeping an eye out for where the game starts to break down, and adjust as necessary when you get there.
And besides, this is a not-uncommon question. After all, to someone coming from any other fantasy game, level 10 seems really, really low! So the motive for asking this is usually just being not quite aware of just how good the level 10 level-up options really are. Take another look at those and consider whether you actually need level 11+ options, or if the game already has you covered.
Best Answer
You can't refuse to level up...
From the Players Handbook, "Experience and levels" p 58:
(emphasis mine) That's not something that you can choose to do or not. As soon as you reach the requirements, you're considered to be of the new level you reached. It's the same thing as aging. You can't decide to remain at 12 years old for six years, then immediately become 18 years old.
Mechanically, that would probably break the game. The players would face encounters increasingly difficult, gaining increasingly more XP. He would potentially gain even more levels than what he expects. That seems pretty unfair to the others players, as they would have to carry him during the time he saves his points.
Furthermore, does he expect to show in front of the BBEG (lv 17, to accommodate the 15-th party level) and then snap his fingers to magically upgrade to level 17 and fight him on equal ground? That doesn't seems fair either, for the players and you.
... unless some very specific circumstances.
The very first entry about creation of magic items and XP cost is located in the Players Handbook, page 88:
So, you can delay your progression, but not choose to stop it, and under specifics circumstances (e.g: being able to craft magic items, and deciding, when you hit the requirements, to craft an item instead). It does not allow you to voluntarily lose a level by spending too much XP, nor does it not allow you to save it. You gain it, you spend it on either, but you don't have a magical pouch of XP somewhere. An "option" to lose a level, is for a wizard to get his familiar killed. The traumatism cause a XP loss, that can be halved on a saving throw (so not the loss is not done voluntarily). You can not "traumatize" yourself by crafting magicals items, as RAW.
Furthermore...
As pointed by Fectin, the section "Experience and levels - Advancing a level" mention a case where you can't access a level:
From what I understand, even in this case, you don't save XP, you merely get a level and the experience needed to get to the following minus one. So you level up. And the additionnal experience is lost. So you're twice as disappointed. No delay, no savings.