As much as you like
It's your money, we don't care how much you bring with you: the more you bring, the more you can spend, we're fine with that.
However there are two important things to note
- You must declare any amount over €10,000 (this applies to the UK as well as the EU, noting that the currency is still Euros when entering the UK! This would be the equivalent of ~£7,800 today, and this has caught people out before who were expecting £10,000 not €10,000). If you don't declare your money, it may be seized and you can be fined up to £5,000 (UK, but there are similar repercussions elswehere). Seriously, I can't stress this enough, declare it.
- You may be expected to explain both where you got the money from, and what you intend to do with it, if the amount is unusual for the purposes of your trip.
Note specifically that the €10,000 limit applies to a combination of all currencies you bring. eg if you have €5,000, £2,400, and $2,300, that would total over €10,000. Equally if you bring $11,400, that alone is over €10,000. Keep a close eye on the exchange rate in the run up to your trip if there's any chance of you getting close to the limit!
Generally, bringing more than a few thousand in cash is likely to be viewed somewhat suspiciously. You'll likely be allowed through customs with it if you have a reasonable explanation (eg if you're a family of 8 and the father brings €10,000 to pay for a 2 week holiday), but you'll draw far more attention than you need or want.
Unless you absolutely need hard currency, consider bringing travellers cheques and credit/debit cards instead, for the bulk of your funds. It's worth having a few days funds on you in cash, but there's little reason to bring thousands of pounds/euros to the UK/EU.
Note that the €10,000 limit is per person, but I'd suggest that if you have more than this as a group, you declare it anyway to avoid any misunderstandings. Declaring just says "Hi, Customs, I'm being open and honest about this", so that they can ask any questions they may have.
With customs: if in doubt, declare it!
UK Reference, of which much is based on EU law. Please double-check my information, though, as I can't guarantee any of it - particularly for non-UK destinations.