http://www.fanuriotimetracking.com/
Fanurio is fantastic in lots of ways. It's only $60 one time fee for a license and then you can use it forever. No web app though, it's for Windows/Linux/Mac.
The timer is extremely well done and easy to use, and it's extremely flexible. You can even have timers running for multiple clients at once. Not only can you toggle on/off but you can pause and easily retroactively add hours. You could have an Android app just to record time and add it when you get back.
Invoice tracking as well as billable/non-billable hours are built as well as customizable rounding (I have it set to round up to 15 minutes). It very clearly shows who has and who hasn't paid and can generate custom PDFs with your business logo on it. It can even track business expenses and mileage as well as keep track of all client info.
You can sync two computers with dropbox, here's a page form the help file on that: http://www.fanuriotimetracking.com/files/releases/3.0.1/help/html//userdata.html
If you need any help with it let me know - I love this software, it's made my day to day life so much easier. I too hated most time tracking apps until I found Fanurio. It's a small team based in Romania and I've emailed support a number of times and they're fast to respond and it's most likely a developer responding to your questions.
Fanurio http://www.fanuriotimetracking.com/ is a time tracking and billing software application, highly appreciated by small teams for its intuitive interface, flexible timer and ability to produce detailed invoices.
Excel spreadsheets work well.
Access is good for this type of thing, but it can be a pain in the ass to learn on the fly. You could google around for pre-made Access form templates to speed up the process.
Fanurio is also a good option for time tracking/invoicing. Single user license is around $60.
If you don't mind web-based, Highrise is also a good option. It's free to under 250 clients. No invoicing included, though.
Several of the tools listed support invoicing. Others may offer the ability to export time logs in a format that can pulled into another software.
Fanurio, what I use, can do both. It isn't the cheapest option but it serves my purposes well and I've had good experience with their support. It's been more than worth the cost to me.