If I subscribe to the many worlds theory, every time I buy a lottery ticket one of me wins.

Yes, but you're not applying the hypothesis to the fullest.
If it's correct, and the number of worlds is infinite, then some of you buy tickets even when you don't. And they win. So, you don't actually need to make the move at all. 😎
- If it's truly an infinite number of worlds, in some of them you win the lottery without even buying a ticket.
  
  Yes. In some, you're given enough wealth that it renders lotteries obsolete.
- In a literal sense, assuming the theory that consciousness in some way depends on quantum processes is correct, this is the proper interpretation.
  Lottery balls being picked seems very unlikely to be dependent on a superposition.
  But (a) choosing to buy a ticket, and (b) what numbers you choose both plausibly could if the above assumption is correct.
  So not only would other yous be buying tickets in other worlds, they'd be buying many different numbers in many different worlds, even if the you in this world wasn't buying any tickets at all.
  And even if the you in this world was now so strongly against the lottery that no future 'branch' of you would ever buy a ticket regardless of the degree to which a superposition might influence your decisions, the many yous from childhood would be so variably influenced in different ways from others around you from birth to now that there might be other parallel yous who superstitiously buy every ticket.
  Even in terms of number selection - if the you here might choose the birthdate of a spouse or children as the numbers, yous in other worlds might have different spouses or children to choose numbers based on.
  Many worlds is a rather boring theory unless also entertaining it with the notion that - like how birds navigate - our decision making somehow depends on quantum effects.
- I just need to invent a device to merge with them.
  
  You already did
  
  Plenty of lottery winners end more miserably than they were before. If anything, invent the device that's gonna make you happy just where you are...

It's possible to have an infinite number of universes where you win the lottery in none of them. It's a common misconception that infinity=every combination when that's not necessarily the case (there are infinite values between 1 and 2 for example, but none of those are 3)
- It's also a common misconception that Everett's many worlds involves an infinite number of universes.
  And that it involves multiple outcomes for macro objects like lottery balls.
  It only means multiple 'worlds' specifically for quantum outcomes, so in OP's case their winning or not winning the lottery would need to be dependent on a superposition of quanta (i.e. Schrodinger's lottery ticket).
  And given the prevailing thinking is that there's a finite number of quanta in the universe, there cannot be an infinite number of parallel worlds. (There could only be an infinite number of aggregate worlds if time is infinite and there's perpetual quantum 'foam' in its final state perpetuating multiple possibilities).
  The theory is much less interesting than is often depicted in mass media (though as of recently is a fair bit more interesting given the way many worlds as a theory would mirror what backpropagation of the physical universe might look like).
- I thought many worlds meant every single possible divergent quantum thingy gets its own universe. There is a universe where a single potassium atom in a banana in my kitchen doesn't decay and there is another universe where that same potassium atom does decay. Multiply that for every single particle in the universe, right?
  I guess even if that was true on a macroscopic level that's not going to guarantee that every possible thing happens?

That's only if you assume that you winning the lottery falls within the infinite, but bounded, realm of random fluctuations between when you bought the ticket and the winning numbers are drawn. There's still physical constraints that the random quantum fluctuations fall within.
An example is, there are infinite numbers between 1 and 2, there's 1.1, 1.11, 1.111, etc. Because of the constraints however, we can still know that none of those infinite numbers between 1 and 2 are equal to 3. Infinite doesn't mean anything is possible.

Serious question: Can somebody explain to me, if an infinite number of universes exist, why do we assume that every possibility must exist within the set? Like, why can’t it be an infinite number of universes in which OP does not win the lottery?
- Your intuition is correct here. OP is wrong. An infinite set of branches of the wavefunction does not necessarily imply that everything you can imagine must happen somewhere in that wavefunction.
  
  But surely the chance of "me" winning the lottery at least once is 1 if I play an infinite number of times
- Fun fact: you can have multiple sets of infinities and even though all are infinite, that does not mean they are all equal. See Georg Cantor.
  
  My infinity is bigger than yours
  
  If you say so :)
- Murphy's Law (edited a bit by the Nolan brothers)- If something can happen, it will happen. On the scale of infinity, this is particularly inevitable.
  
  But what evidence is there for this being true?

While the basic idea is interesting, the statement is misconceived. It confuses what you believe to be possible with what is possible according to quantum physics.
For your statement to be true, the lottery would have to be set up in such a way that the choice of winning lottery number is decided by the outcome of a quantum measurement which includes the possibility of your number being chosen. The outcome would then exist in superposition, and as soon as you learn the result, you are entangled with it and enter into superposition as well.
But like I said, the core idea is still fun to think about, because this type of branching happens constantly and it becomes an interesting philosophical dilemma of how to think about what could possibly happen, not merely what does (as far as any 'you' can tell). Imagine if you could experience all outcomes of some particular chain of events and how that would affect the way you make decisions.
- Like the idea of winning is more fun than actually winning. Until you check you have both won and not won.

Even if you don't buy a lottery ticket an infinite of you wins

However, every time you buy a lottery ticket, almost all of you lose a little bit of money.

An infinite number of you wins.
- Someday that me might be me.

In at least one world, the ticket machine short circuits and electrocutes you. In one of those, you become a superhero named Lottery Lightning.

In many of those worlds the lottery doesn't exist.
In many of those worlds you don't exist.
- The possibilities are endless.

But also, every time you buy a lottery ticket, one of you gets a paper cut from it and dies from an infection from antibiotic resistant bacteria.

Correction: at least one of you wins.
It's possible to buy a lottery ticket where ALL of the alternative universes wins the lottery EXEPT you

Also, everytime you buy a lottery ticket you also don't.
Likely one of the "other" yous won when you decided not to buy a ticket.
- So a version of me must win every single lottery, but I will never win any of them if I don't buy a ticket.
  Gotcha.

Yet you'll go bankrupt 10 years after in every one you win. This universal constant defies the multiverse.

You'd have to pick numbers purely randomly. Maybe construct a machine that randomly decays cesium. If it does, it triggers some sort of mechanism to register a 1 or 0. Maybe put it in a box with a toxic fial and a cat. When it triggers, the toxic substance is released and cat dies. So depending on if cat is dead or alive you get 1 or 0. Generate enough bits this way and you'll get numbers to put in the lottery.
- How many dead cats would that take?
  
  It's impossible to tell - it's random. But on average it would take a number of cats equal to half the bitsize of the numbers you generate.

And every time you don't win, you can slowly confirm your darkest fears that you are indeed in your shittiest timeline.