Is the American standard gauge the largest (widest) gauge in the world for rail transportation?
No it isn't. Here in Australia the originally agreed "standard gauge" was the 5ft 3inch (1600mm) Irish Broad Gauge. Four of the 6 states started building in this gauge but only one persevered with it. My state quickly changed to Stephenson Standard Gauge (4ft 8.5in - 1435mm and this is the gauge used in the USA and Canada) and that ultimately became our national Standard Gauge. Two states still use the Irish Broad Gauge but one of those two states also has rail lines built in Standard Gauge and Cape Narrow Gauge (3ft 6in - 1067mm), which was an enormous disadvantage to transport, trade and commerce as both freight and passengers had to be moved from one train to another at various places around the state. That situation still exists but to a much lesser extent as there is now a standard gauge rail network connecting all mainland state capitals.
In Russia, Ukraine, and other parts of Eastern Europe the "standard gauge" is 5ft (1520mm).
The widest commonly used rail gauge is the Iberian Gauge used in Spain and Portugal, at 5ft 5.5in (1668mm) but they also have trains with variable gauge wheel bogies. These trains can be driven through a machine that in a few minutes will unlock the wheels, move them inwards or outwards along the axle, and then lock them in the new position. So an Iberian gauge train can be converted to or from standard gauge without first unloading freight or passengers.
Addendum: The Great Western Railway in the UK built its lines with a 7ft (2134mm) gauge, which was later widened slightly to 7ft 1/4in (2140mm). But by 1892 all their lines were converted to Stephenson Standard Gauge.
Wider gauge track does have a higher stability than narrower gauges but it also has higher construction costs for bridges, tunnels, cuttings, etc. It also takes up more space because curves need a larger radius. Cape Narrow was often preferred in the 1800s because of its cheaper construction costs.
