Before submitting this question, I had one final go at searching. And lo and behold! TV Tropes has an entry about this idea (warning: don't click that link unless you have half a day to spare). One of the examples is a short story by Mack Reynolds, called "Compounded Interest" which is indeed what I was looking for.
[...] a man in medieval Italy, depositing a modest sum in pure gold with a bank, for a period of a hundred years. He also gives some predictions and directions on how to invest it.
Matches. The year of the first appearance of mister Smith is later dated as the year 1300. The story opens with:
The stranger said in miserable Italian, "I wish to see Sior Marin Goldini on business."
The sum is indeed modest, but is deposited for a hundred years:
Marin Goldini said impatiently, "Sior, the amount is hardly sufficient for my house to bother with. The bookkeeping alone—"
The stranger broke in. "Don't misunderstand. I realize the sum is small. However, I would ask but ten per cent, and would not call for an accounting for... for one hundred years."
A century later, a similar man appears, not withdrawing his accumulated wealth, but reinvesting it, strengthening the contract, and again making predictions.
Mister Smith held out a torn sheet of paper. His Italian was abominable. "The agreement made with Marin Goldini, exactly one century ago."
"[...] I am going to request you have drawn up a contract more binding than the largely verbal one made with the founders of your house."
This goes on until the 20th century. It is by then the largest fortune in the world by far, unbeknownst to all but the select few managing it.
"[...] He is scheduled for July 16, 1960. At that time, friends, I am of the opinion that we shall find what our Mr. Smith has in mind to do with the greatest fortune the world has ever seen."
Von Borman looked about him and growled, "Has it occurred to you that we eight men are the only persons in the world who even know The Contract exists?" He touched his chest. "In Germany, not even the Kaiser knows that I directly own—in the name of The Contract, of course—or control possibly two thirds of the corporate wealth of the Reich."
I think it's even hinted at that both World Wars were fought because of the fortune.
Strongly hinted at the First, the Second isn't mentioned.
"This time Mr. Smith was wrong," Borman growled. "As he said, oil is to be invested in above all, and how can Germany secure oil without access to the East? My plans will succeed and the cause of The Contract will thus be forwarded."
The quiet Hideka Mitsuki murmured, "When Mr. Smith first invested his pieces of gold I wonder if he realized the day would come when the different branches of his fortune would plan and carry out international conflicts in the name of The Contract?"
The man appears and wants to liquidate his assets, because he needs the funds to build a time machine and go back to medieval Italy to deposit the gold that started it.
Mr. Smith looked at him. "I wish to begin immediate steps to liquidate."
"Liquidate!" six voices ejaculated.
"I want cash, gentlemen," Smith said definitely. "As fast as it can be accomplished, I want my property converted into cash."
"[...] I have finally solved, I believe, all the problems involved. I need now only a fantastic amount of power to activate my device. Given such an amount of power, somewhat more than is at present produced on the entire globe, I believe I shall be able to travel in time."
I most likely read this in SF — The Best of the Best, edited by Judith Merril.
The twist I hadn't remembered was that
the time traveller was a scientist contacted by the bankers in charge of the fortune, who were investigating the possibility of time travel to explain the strange appearances. That triggered the scientist to actually do it.