Price range €300-€400
Battery life with normal use > 1 day

My current phone is a samsung galaxy gio with android 2.3, a scratched up screen and a battery life of 6-7 hours with minimal use.
The new phone is supposed to be the opposite of this old phone. I want to use it as a digital agenda for school, so it needs to have a nice large screen so I can fit all appointments on there.

As for Apple, you'll pay for the Apple logo

Sure. (Although other comparable smartphones aren't very cheap neither)

and a locked down device

Sure again, until you jailbreak it if you ever need whatever can be offered through jailbreaking (btw iOS7 has some very nice additions whose ideas probably came from the jailbreaking community)
I was myself using a jailbroken iPhone 3GS for 3 years and the few things I was actually using it for was SBSettings (quick settings toggle, like wifi, etc.), removing system icons I don't use, and using BiteSMS for quick-replies. Now, I only miss BiteSMS's quick-reply, which was just a few-seconds-time-saver, since the rest is already there.

that you'll have to reformat every few week.

Wait.... What ?
The aforementioned iPhone 3GS had only been restored 2 times. In 3 years.
Why would you want/need a higher formatting frequency ?

Also their touchscreen is out of sync with the content, which makes it incredibly hard to type text.

Wait... We clearly aren't talking about Apple i* devices anymore, here, right ?
Among the smartphones I have tested, the actual touch interface is, to me, the best.
Other manufacturers don't seem to be this precise, but do have a better screen quality in terms of contrast, brightness etc (with super amoled and stuff). But that's when you really want to get technical anyway. For 99% of people, A standard "Retina"-like screen, amoled or not, is perfect.


Anyway, the OP wants a "nice large screen" so probably some recent Samsung/HTC stuff ?

I have the Samsung Galaxy Ace 2 and it's pretty good. Great battery life too.

Adriweb I had to reformat my iPod Touch 4 three times because it crashed beyond possible recovery, and two of these times were before it was jailbreaked. Although it crashed much less often than Android, when it did it was really bad. I have never been forced to reformat my Android phone yet.

As for out of sync screen, almost everytime I typed a letter, especially the ones on the edges of the keypad, it instead typed the letter next to it. The worst part, though, was clicking links: In order to access a link without clicking the wrong one, I had to zoom in until the font height takes almost half of the vertical screen height. Also, one of the most annoying thing was how the screen kept dezooming by itself when scrolling. I hardly ever had any of those problems with my Samsung Galaxy i551.

Yeah the GS3 is pretty good too but the N4 outperforms it while being cheaper for a brand new one. Not to mention that it has a pretty active ROM scene.

Never had battery issues with it. But you're right about that. (though you can find external batteries that work with anything)

If you really want battery life, you might want to consider buying a normal phone instaed of a smartphone. I am currently using a nokia C2-01, and it does the job very well for me. It's controlled by normal buttons instead of a touch screen (which also means that the screen is smaller than on smartphones), but the screen size and resolution isn't that bad. Also, the phone is very sturdy. I have dropped it several times, and there are no cracks or scratches in the screen. It has also got a few built-in apps, and most of them are usefull (agenda, a thing to save notes, a timer, an MP3 player, a video player, and a few games including snake). The video player doesn't support a lot of formats, though.
It also has got a reasonable camera (3.2 megapixels), and a battery life of almost 2 weeks with minimal use and 2-3 days when you use it (and especially it's camera, which seems to use a lot of power) a lot.
So if you're only goning to use it as a cellphone and agenda, I would suggest this phone or a simular one. With a price of ~€80, it's well below your budget as well. However, if you are looking for a mobile gaming and internet device, then it is less suitable for you, since this phone has no wifi and not a lot of games are made for it.

Edit: Both me (who owns this phone) and my friend (who owns a Samsung Galaxy Mini) agree that this cellphone is more usefull for school stuff.

In any case if you need a phone that lets you type stuff frequently (eg the agenda stuff) and requiring you to click on smaller stuff, then a smartphone with a larger screen is a must. You might want one that fits in your pocket, though, so it's safer from thieves.