The Fashoda incident could turn into a World War, with different leadership in Britain and France. Either of the Moroccan Crisis could turn into a World War.
Maybe the Balkan Wars could be expanded?
There's always the possibility of some other "damn fool thing in the Balkans".
This is pre-1900, but what if Germany and the US go to war in 1898? Germany wanted the Philippines, and were acting rather aggressively towards the US navy during the Spanish-American war. The US threatened to fire on the German vessels if they didn't play nice and Germany backed down. As far as I can tell, the reason was that the Germans figured the US would lose the islands to rebellion within a year or so, leaving it for them anyway. I'm not sure what the alliance systems were like at the time, but it seems like the kind of thing that could grow bigger under the right conditions.
Russia would have to be careful here not to alienate the French who were the "protectors" of the Roman Catholics in the Empire. Moreover, I would not see an naval arms race here because armies are going to win this war.There are Russian ambitions against the Ottoman Empire that could lead to a blow-up since they are both plunged into a new naval race