Are there advantages to treble nationality that are specific to a travel site?
Yes, if you travel a lot to places that require registration or visas, then this question is topical without the need for an exact enumeration of which nationalities are concerned.
By and large it can open more places to visit without needing a visa. Taking my own case as an example, I can get in to about 19 different time zones without a visa. Not every country in those 19 time zones mind you, but at least one. A big thrill is that my children qualify for the Naval Academy as university in three different countries because they also hold those nationalities.
Not in my case, but depending upon where your treble nationalities are, you may be violating a local law about having multiple nationalities.
The big downside is that one of those countries has some awkward tax issues that arise when a national is in a 'foreign' country, and this tends to limit how open someone can be about their nationalities.
I have already served in the Navy of one of those countries and am not eligible for conscription, so I can bypass those issues (if there are any).
There is also a lot of extra paperwork involved in tax reporting (for those countries that require it).
Another downside is that those passports expire and it's expensive keeping them up-to-date. I qualify for a 4th passport, but have opted not to pursue it because  it does not add to the places I can get to; and  the on-going expense of passport applications; and  I could not pass the nationality to my children.
I expect at some point the Department of Homeland Security will start some programme that makes it awkward to have passports for countries they deem to be inimical to their own interests, and that would be another disadvantage. The US, for example, always adds an extra layer of inquiry whenever I arrive there; I expect this to get more intrusive as time goes on. It's not that I have something to hide, but it's an unwelcome consumption of time.
My overall advice to people in the same situation is that if they can pass their additional nationality to their children or if it significantly broadens their visa-free access they should pursue the paperwork and get it done. I have also advised Americans and Canadians to carefully check if there's a (for example) Lithuanian in their family tree.