I want to take a TGV for the experience of traveling very fast on the ground. The Lyria trains are advertised to travel at 524+ km/hr. The TGV I took only went 300 km/hr. Does anyone know which routes travel at full speed, preferably one that connects with Paris?
There is some confusion, top commercial speed for TGV is 320 km/hr between Paris and Lorraine (on the way to Metz, Nancy, Strasbourg, Luxembourg, or Germany). Some conventional high-speed trains run or ran at 350 in Spain and China, 400-430 with Maglev technology, but not at 500 and more. Some parts of the French high-speed network were also designed to support 350 or 360 but to my knowledge no trains have ever been scheduled for commercial service at these speeds in France.
I don't know where the 524 figure comes from, but various TGV trains have held the world speed record for conventional trains at 408.4 (1988), 515.3 (1990), and finally 574.8 (2007). Those were achieved with slightly modified but mostly standard trainsets so the TGV is capable of higher speeds than 300 but you can't experience them anywhere at the moment.
In fact, the train is not the only thing that must be modified to reach record speeds, the overhead line are also adjusted so that each record attempt corresponds (not coincidentally) to the end of the construction of a new high-speed line in France as it is the only time when a new series of test runs can be organised without unduly disrupting operations.
As an addendum to Relaxed's excellent answer, there's only one place on the planet right now where the public can experience traveling at 500+ km/h on a train, and that's JR Central's Yamanashi maglev test track in Japan.
However, test runs open to the public are scheduled only irregularly, and are extremely popular. The last runs for 2,400 lucky winners were in November 2014, and there were over 300,000 applicants.
On the upside, the current test track is being extended into the new Chuo Shinkansen maglev service between Tokyo and Nagoya, which will operate with L0 trainsets at up to 505 km/h. (This train is also the current holder of the world speed record, 603 km/h.) While the full line is not scheduled to open until 2027, JR Central is looking into running partial service during the 2020 Tokyo Olympics.
From one of my professor, in a French transportation engineering school, the main reason for TGV in France "only" going at 320 km/h are security and comfort standards.
In fact, for security, the main reason is that TGV are attended to be able to stop within, if i'm not mistaken, like 3350 meters or so (order value). driving at more than 500km/h means so that this stop distance will increase a lot with current systems, thus not meeting standards.
Regarding comfort, most of the turns on TGV tracks are calculated to not provide too much discomfort for passengers (I think it's about felt acceleration), and going fast will increase this discomfort.
Finally, yes this is possible with a TGV to go at 500km/h, but the whole current systems (TGV and their track) need to be changed, and that cost too much actually (and I did not mention other aspects like wear and tear, electricity costs and so on).