Working for a Germany company, I am surprised that you are included in this. Normally, trainees should be exempt from anything that costs private money, because everyone knows they don't earn much money.
With 200 people, you are basically guaranteed a free breakfast every day. You could easily be eating as much cake all year as you bring yourself once a year. Now your budget does not account for eating cake every day, but it's not as much cost as the plain bill could make you think. You get a lot in return. As with all expenditures, plan in advance and save some money.
You could ask your direct supervisor, who will know exactly that you cannot afford that kind of money how you should handle this. Chances are he will realize this is a hefty cut in an Azubis'1 paycheck and make room for an exception.
If possible, you could find somebody close to your birthday and split the bill with him, doing a double birthday celebration. If you both bring 60% of the cost, 120% will be more cake than anybody can eat.
Another option is to find a cheaper alternative. Maybe if your birthday is in the summer, Icecream for 200 people bought in bulk might be cheaper than cake.
Or bring a waffle iron and make waffles there. Ingredients are dirt cheap, but having a waffle fresh off the waffle iron goes a long way to make it memorable.
Alternatively, there is a lot you can do to make it cheaper.
- Don't buy cake, make it yourself.
- Put up some nice decorations
If you make the cake yourself, you can easily come up with about 100€ for 200 people. It does take a lot of time though. It does not take as much skill as people would like you to believe. Grab an easy recipe and try it a few times before the big day. Personally, I do Brownies. Take your time on the weekend and you should be set.
Nice decorations are cheap and don't feed hungry colleagues, but they do show that you care. It's a completely different feeling if you are going for a second piece and there is an empty box on a nicely decorated table versus an empty box on a plain table. You cannot really be angry at someone who cared. After all it's not your fault it's all gone, you did not eat it, they did.
And as a last resort... you could take the week off. Maybe they'll forget.
1 - Short for Auszubildender, an apprentice or trainee.