You possibly could. You'd have to get a driver who would agree to do that, many drivers will not. Looking at the Uber page for Memphis it shows the charge is only $0.15/minute so a driver waiting for you would be costing $9/hr plus $1 a mile. If you have a lot of driving someone may do it, but if it's a lot of waiting, I don't think many drivers are willing to wait for you at only $9/hr.
How does it work? You request a ride, and when the driver gets there tell them what you have in mind and ask them politely if they'd be willing to do that. When they tell you no you accept that because they drive Uber to earn money, not serve you. And by waiting on you all day they make no money. If they say yes, you are very grateful and have at least $50 cash in your pocket to tip them with at the end of the day. Preferably more.
My suggestion is that when you get your first ride of the day, ask them if they are willing to do it for the day, you can't really prearrange this unless you approach drivers in advance. You will probably have a hard time because the waiting time billed is so low. The 9. an hour and $1 a mile Renaldow mentions is the gross, the driver only gets 80% of that. So I don't think you are going to get someone to wait for $7.20 an hour. If the convenience of having someone at your beck and call is worth it to you I suggest you approach the driver and offer them at least $50 above the fare, promise them that up front then they might be willing.
If it's insisted on having the same one all day, then do the math and pre-arrange an hourly figure to tip for waiting time. Per some of the posts above it's already known that the driver is making $7.20, after Uber's cut, per waiting hour - meaning no driving. So one could say "I'll tip you $10 per waiting hour". Give the first hour up-front, and the following tips at the end of each hour completed. If you actually get driven around during an hour, tip an extra $5 for the ride itself. So some hour's the tip would be $15 and for hours the driver just wait's they'd get $10. Of course my numbers are purely opinion and just used for example...adjust them based on personal preference and whatever the driver agrees to. That's really the only way I see it working unless a sucker driver gets reeled in to this...in which case that'd be a great deal - for you.
Assuming you will be in areas that are fairly traffic dense, it would be cheaper for you and not much less convenient to request a new car everytime you need it. In most areas of a city like Memphis, 10 minutes is probably your maximum wait time, so its not going to take a ton of extra time and will save you the cash you would be dropping on idle time
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