This page links to examples of data that is as current as technically possible. If you know of similar sources, please list them here.
Open access to real-time satellite data is en emerging interest area (see recent Nature article): check out Landsat for imagery and data. Learn about using OpenStreetMap's data which can be downloaded historically, or looked up in hourly or minutely differentials as people edit the world.
The location of Swiss public transport in transport_api can be used to predict the position at any given time. This resource was built and is maintained by the Opendata.ch community. Meanwhile from the Geneva public transport open API you can get measured locations and arrivals.
All large Internet communities generate tons of data and can be studied in close to real-time. For example, here is a visualization of live Wikipedia edits. You can get historical wiki data from Stanford and Amazon. However, most other data is commercial and licensed for profit.
The Internet of Things is a movement to put all kinds of devices online, it is important that their data is accessible with high precision in time. You can find public data feeds on these sites: http://thingspeak.com/channels/public, http://data.sparkfun.com, http://thingful.net, http://dweet.io/see.
Also look for crowdsourcing projects like Blitzortung (real time lightning strikes) which can complement existing - and usually nationally-bounded - data sources like US water data.