If you know the exact results (like Biff in Back to the Future), winning things like lotteries or horse race results multiple times will draw attention.

Actually, any kind of money hoarding - moreso than anyone else - will draw the attention. Which might be OK, of course, but if you make waves that are too big, you'll affect global policy and the course of (economic) history, and all your predictions may be off.

But, getting significant but non-controlling stakes in various successful stocks might work. You'd be like Warren Buffet, who I consider to be rich, but otherwise anonymous.

In other words you end up in manure anyway.

I actually loved the side-plot in "About Time" where the main character went back in time and changed something and then they had a different child and then had to undo their change.

Create life and casualty insurance!

Illegal in that period. Something like shipping insurance can be built from a repurchase agreement and this may get around usury/gambling laws, but how do you do that for life insurance? That takes more legal than probability knowledge. Also do you have the skills to combat insurance fraud?

Annuities are the same math but different product, and got valued first by Haley. I'm not aware of life insurance being illegal per se, and there might be ways to work around it via structuring as a fraternal organization ala the origins of the insurance industry in the US.