Of fun things to do, I will mention some.
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Shrek Adventure
There are themed attractions such as the Dungeon ( not tremendous ), the Aquarium, the London Eye, etc within or beside the old County Hall, located by the south side of Westminster Bridge, as you're probably aware,
A recent addition very close to the London Dungeon, within County Hall building, is the
Shrek Adventure. If you into the Shrek films and the humour of them, then do go, because this the best of the bunch in London. The design and props are by Dreamworks and the staff seem better and more into their roles than at similar. The quality of this experience is up there with what you'd get say at Universal. It is a laugh. You don't get to see the actual Shrek much which surprised me but still, you are in the Skrek world totally, going from place/scene to the next where you meet new actors conducting their bit of the experience. Near the beginning there is a "4D" London bus ride, driven by Donkey. There's even a sing-a-long before you board !
This won't appeal to everyone of course but if you are into the Shrek thing, this is the best attraction of its kind in London.
Entry is by a
timed slot, which you choose when you book online. Don't don't it any other way. A word about the other attractions you may find bundled with Skrek to save money - Don't ! If you're going on the other ones buy single tickets direct or face unexpected hassle on your short trip, trust me.
Shrek's Adventure! London | Official Tickets and Offers
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The Shard
The View From The Shard, staying at the Shangri-La or not, is definitely worthwhile for it's amazingness, differentness, bizarreness ! Book a timed entry slot.
Think about the sun position and what will be illuminated below. Me, I went up near tea time, because the light is nice then and by the time, it was shining eastward, upon the Tower Bridge, Waterloo station with its "toy trains" moving about, and eastward over the interesting dockland areas and where the river snakes a lot.
The View From The Shard
Very close, is the interesting and fun Borough Market, where you can snack and drink in this old Victorian iron gem.
Since you're in the area you might as well see this early operating theatre.
Old Operating Theatre Museum and Herb Garret Web Site
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Speakers Corner
I cannot fail to mention this whenever I talk about London trips. Since you will there on a weekend and it is still the summer, Speakers Corner is open to you.
Hyde Park, over from Marble Arch ( north east corner of the park ), on
Sundays only for a good selection. Gets going well in the afternoon
Get to Marble Arch on the Central Line if coming by tube and use the well signed underpass, to emerge on the park side at Speakers Corner.
While the speakers in this modern era are mainly not great, it is hilarious as an entertainment. Nowadays we tend to see extremists and religious nutters, but that's the fun. Yet there are also a few good speakers.
By mid afternoon its teeming with speakers. Personally, I like to spend a whole afternoon here, but even if you go and have say an hour and a half there, you'll get the experience. If you haven't done this thing before it will be a memorable experience for you.
There is a snacks & drinks kiosk by the speakers and ice cream.
Speakers Corner
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Flight Simulators at the Emirates Aviation Experience
Now obviously, this will appeal to some, not others. Are you aviation lovers ? Are you geeks ( and proudly so ) ?
This is forty something quid - very reasonable considering. You get an experience of realistic aviation simulators. Choose 777 or Airbus 380 . They are set within the fairly interesting Aviation Experience. The sims are static, so don't expect a big pod that bobs around on hydraulic legs, but still, this is really good, especially for the money and a huge step up from what you could do on a home rig.
Flight Simulators | Emirates Aviation Experience
Book, obviously.
The location of this is actually at the southern end of cable car thingie over the Thames - another fun thing to do.
The Emirates Air Line
You can pay for it with your Oyster by the way.
And that reminds me, send off for an Oyster. Keep it and use it on subsequent visits.