Cheapest i5 or i7 with ssd machine

this looks like the cheapest possibly sane thing on Amazon (i.e. not refurbished or older generation stuff).
or you could throw ram and an ssd in a barebones like a Gigabyte brix, but I'm guessing you probably want a desktop i5 or i7.

Absolutely not an endorsement from me for the link as I've never heard of them but it meets the stated requirements and budget (depending how much windows costs you) without having to build it and you might find others out there.

For a basic internet PC you can go really cheap these days,

Motherboard €55.44
CPU €54.47
SSD €48.85
Ram €41.45
PSU €32.85
Case €40(ish)

Plus €30 for postage from Germany is €264.61

Windows. You can buy a key from Reddit for around $35 or from Microsoft €95. For a basic internet PC try Ubuntu Mate and see if it works for you. It operates like windows but without much of the maleware.

If you want to build for yourself your budget is easy and it's just about what (if anything) you throw some extra money at e.g.
i5-6400 to i5-6600 (go 6500 or 6600)
120GB to 480GB ssd (go 240GB or 480GB)
4GB to 16GB ram, dual channel or just one dimm (go 2*4GB)
budget psu or something a little more future proof (I'd go to e.g. Corsair CX430 which will let you add a gpu later if you want, they aren't only for gaming, though it might only come back to you on the resale value)
cases you can spend anything you want on (ish) but something like an Antec NSK3100 is a little nice but still pretty cheap
Windows ... well there's cheaper and dearer ways let alone Home or Pro (Home should be fine) ... or Linux

So you could do expensive windows, i5-6600, 240GB ssd, 2*4GB, CX430 and NSK3100 and expensive Win 10 Home (usb stick) and should get in under €600 or close enough and have something very nice and pretty future proof. Drop the cpu to a 6500 if you can't quite get in under the line (or go a cheaper route for Windows).

If you explain a little more about what else you do, or might do, other then surf the net, someone might refine the choices down a little more for you.

Taking windows out of the equation on how to get the most out of 600, my main question is how much storage space you need and how much ram you want/need. As you are generally focusing on "most powerful" I'm going to suggest a 250GBish ssd should be enough for you and that should let you think about an i7-6700 based machine with your 600 euro (though see below, I suspect you are better off sticking to the i5 and might even get a better machine that way despite it not quite having as much raw cpu power).

€35 Antec nsk3100 (amazon thanks to mindfactory shipping)
€45 Corsair CX430 (mindfactory)
€56 MSI H110M PRO-VH (mindfactory)
€309 i7-6700 (mindfactory)
€80 16GB (2*8GB) 16GB Crucial Ballistix Sport (mindfactory)
€75 Samsung 250GB 750 EVO (mindfactory)

That should be exactly €600 plus shipping give or take a couple of euro. You could use a cheaper SSD (e.g. 240GB Silicon Power for €60 or a Sandisk for €64) to maybe get it under €600 shipped (and/or drop back to 2*4GB of ram) but I have a preference for Samsung, Intel, Toshiba SSDs on the reliability stakes (which may be out-dated at this point or not worth the premium but I change those opinions slowly for stuff I have to care about myself).

If you dropped back to the i5-6600 you would have another €110 to play with and to be honest the i7 is a hefty price premium, so unless you have something you know needs every last drop of cpu performance (in which case it's probably worth mentioning) you would likely be better off saving that. If you do you should probably put some of the saving into moving up to a bigger SSD just to be sure to be sure (e.g. €124 for a 500GB Samsung 750 EVO down to €100 for a 480GB Transcend SSD220) and pocket whatever you have left over from the €600 or put it towards windows or a DVD writer (about €14) or a fancy cpu cooler if you want to try and get it quieter or a new keyboard/mouse or a card-reader or any other bits you fancy.

Really €600 isn't quite enough to do an i7-6700 full justice. I'd rather pair it with a motherboard that would cost about 3 times the price with thunderbolt, usb-c, displayport along with hdmi so you could go for dual monitors or 4K properly), m.2 pci-e for an insane speed SSD, 4 Dimm slots, more fast pci-e slots (i.e. SLI capable) and overclocking ability, i.e. a €169 Gigabyte GA-Z170X- UltraGaming.

If you want to (and can) stretch the budget a little you could also go for that board with the i5 with a €91 256GB Intel 600p m.2 blazing fast SSD and say a €48 Antec GX500 as it's a bigger board that needs a full atx case (I'd have said nsk4100 but it's out of stock on Amazon right now). That should be about €645 + shipping (or maybe €615 with "only" 2*4GB of ram with the free slots to just add another pair later). Something like http://uk.pcpartpicker.com/list/YZNsnn (which it says is about £580 or €680).

Hopefully know you realise the exact budget wishes and hardware demands can help tweak down the recommendations you get but this you should hopefully have enough to go on now to try and pick a course.