Church of Christ, Canterbury
There are two or three totally stand-out churches in England and the city of Canterbury hosts one of them. In the middle ages it was the second most visited church in Western Europe after St Peter's Rome. If Henry II thought he was hurting the church in England by assassinating its archbishop all those years ago, he misunderstood the mindset of the pilgrims that flooded the cathedral with cash donations for the hundreds of years to follow. Not that it hurt the English Crown that much financially, as Henry VIII nationalised Canterbury Cathedral several hundred years later and took all of the money and gifts.
One of the consequences of the huge pilgrimage trade, and something Henry VIII could never appropriate, is that the town of Canterbury is full of tiny back streets that curl around the cathedral's walls and end up in clusters at its gates. Another is that the church still doesn't mind hitting the visitor for a sizable donation for entry. At least they have the courtesy to do it at the gate house and not at the church door. If you go within 15 minutes of a service starting, the gates are open to all. If you don't want to go to a service, then wander the extensive gardens until it is finished. Once you are in, you are in.
Within the walls is a huge complex of buildings, a school, hotel, many secret gardens, medieval town wall, administration offices, the Archbishop of Canterbury's Palace and any number of other things. If you stay at the hotel, you get a resident's free entry pass (which is nice) as well as a car park (which is amazing for central Canterbury). It is quite lovely to be locked within the walls at night, able to wander around while the outside world is without.
The building itself is a complex mixture of styles but each bunion (as I believe the Prince of Wales terms "modern" extensions to traditional buildings) seems to be well moulded into the main fabric. For a famous church, many parts are less grand than you might expect, bringing to me, at least, the realisation that this is a working albeit large church in Kent with its own daily issues to contend with (not least things breaking all of the time and needing to be replaced and the fact that ad-hoc solutions can come to stay forever).
Unlike other churches, such as Ely which are journeys through time and within the space in and around them, the Church of Christ in Canterbury is a series of places to explore. It isn't as seasonal as Perugia in Italy or Ely high in its fenland world but instead each room and garden is fixed in time and place - but there are so many it feels as if there is a different one for every day and mood.
There are a good number of services scheduled so if that is what you are after, you should be able to find something to go to. The choir is one of the best I have ever heard and the boys (who were singing when I was there) were as obviously fond of their strict choir master as he was of them. They use the wonderful wooden church within the church - the choir stalls. So evensong is probably a good option for many visitors. They also do very modern styles of church service and they can host up to three concurrent events in the main building (including amplified bands).
Many of the old monastic buildings are still in existence. If you like to potter around very old buildings, the eastern side of the church is the place to go. Once you are in the middle of the old buildings (some in ruins others in use) you may find the "dark passage" into the crypt. Unlike many crypts, this one is actually all above ground: the cathedral above rises to sit a whole storey up above the ground leaving the crypt at ground floor level. It is a wonderful space with low arched roofs supported by a dense array of round columns. Even if the central area gets a bit noisy, there are lots and lots of chapels around the outside, many with tiny doorways. You should be able to find somewhere quiet. It is here that they occasionally hold electronic music events by candle light as no sound ever gets upstairs to interfere with quiet services up there.
The place does get overrun with school children on bus tours from the continent. Canterbury's location is not an historical accident. It has always been the first stop after getting off the boat from France. It made it easy for the bishop to get to London or Paris. The tension between the requirements of London governance and European food and culture has deep historical roots.