Well near us there are loads of free playgrounds - but just as nannynick pointed out, you have to pay to park at most of them.
1 trip to Soft play will be just shy of £10 for 1 adult and 2 children - free parking but a few miles so petrol money. Any of the children's farms will cost between £10 and £15. That's not including any snacks or treats there. So just 2 outings a week could cost £30 easily, a few extra pounds for parking for the free ones.
Any of the set 'classes' usually work out to about £7 of £8 per session, but must be purchased in advance so cost regardless of whether the child participates that week or not, and are either quite a drive away or nearby but with parking at a cost.
A cup of coffee is easily £2 - £3 - and a lot of these places don't make it easy to bring your own lunch (eg separate seating areas which just aren't as nice, etc).
I take my DC to coffee shops for a cake and a cuppa/juice because I want them to learn how to sit (relatively) still and behave themeselves so that I'm not embarrassed when I take them out to functions/events with other people - again that costs money.
'Play dates' are probably the most economical activity - but it takes a lot of time and effort for nannies to develop a network of contacts if they're new, and if they're experienced they will be more expensive already - bit of a catch 22 cost wise.
TBH it's why I use a CM 2 days a week instead of using a part time nanny now. She has her own memberships and takes the children on outings of her choice on those days.