Is there a database of global rainfall time series?
Yes. Nearly global satellite and radar derived rainfall data are hosted by NASA Goddard Space Flight Center, namely:


Tropical Rainfall Measurement Mission (TRMM) aboard the TRMM Microwave Imager (TMI), a multi-channel, dual polarized, conical scanning passive microwave radiometer designed to measure rain rates over a wide swath under the TRMM satellite. TRMM was launched on November 27, 1997 into a semi-equatorial (+/- 35 deg) orbit carrying 5 instruments. Three levels of data processing are available on the NASA's server, where higher level indicates more processing, quality control, and merging of multiple raw datasets. For your purposes, use level 3 TRMM as it combines satellite data from multiple swaths with precipitation radar data and other sources, and provide 3-hourly data in the 60S-60N latitude band, at 0.25 degree horizontal resolution.
Global Precipitation Measurement mission was launched in early 2014, and provides full global coverage at 0.1 degree horizontal resolution at 30 minute intervals in time. This product is in its very early stages, and is still under development and testing. 


In both cases, if you are interested in more precise rainfall estimates at a particular point in space and time, look at level 1 data. It is raw data that provides a time step for every single point along an orbital trajectory of the satellite. Use with caution.

In addition, ground validation data is available on the same server, and some or most of these data is assimilated (merged) into level 3 TRMM and GPM products. 

Once you download the data, you can use their data viewer, THOR, or use your favorite programming language or analysis software to display it yourself.