Set a weekly budget. You could pay a set amount each week and if they wanted to do something costly, they could save up most of a weeks budget and combine it with the following weeks budget.
Have a mileage log for all trips in the car, so you get a feel for how far they are travelling.
For the weekly budget... £5-7 per day may well be sufficient. Consider how much things cost - a trip to a park / woods isn't always cost free (especially if there is an ice cream stand at the park). As you found with your previous nanny, children can get bored doing the same thing all the time. However they also quite like repeating things they found they do like - I find the children I nanny will request repeat visits to museums they have liked, parks/playgrounds.
Some parks are better than others... even your children will say that I expect. I get requests to take the children I care for to the "curlywurly slide park" - it's not our nearest park, it involves a drive, but it has a slide that goes round and round, which is not that common in our area these days.
As nbee says, annual pass for a local themepark can be very handy as if going frequently the cost per visit drops a lot. Our local themepark is designed for under 10's I would say and has great playgrounds inside it, as well as puppet show, stunt diving show (guessed where it is yet?), trains and aquarium.
Set budgets and set expectations. Give them suggestions for local places to go, it's hard to find places when you start working in a new area so you tend to go back to places you have been in the past.
How old is your eldest? They may be at an age where they want to see things like Castles, Aeroplanes, Trains, Dinosaurs... whatever takes their fancy. If you don't fancy doing those sorts of trips out at the weekend yourself, then having your nanny do it during the week saves you the trouble of doing something you don't want to do. Not saying you don't want to do those things at weekends but some parents are not keen on going round museums, but your nanny might be.
Does your nanny absorb any of the cost themselves? As a nanny I tend to work to a budget but if we exceed the budget it comes out of my money. I pay for my own themepark annual pass, parents pay for the children's. If going on a long car journey, I don't claim full mileage - only claim 1/3, which I feel represents the petrol I use. Some nannies (and parents) may feel that nannies should be claiming all costs incurred... however over the years of working for various famillies I have found that if I did that, we wouldn't do as many things, wouldn't get to explore the country we live in.