The Time Traveler's Guide to Financial Security

So you've invented a time machine. First, let me offer you my sincerest congratulations; that is quite the accomplishment! Now you're interested in monetizing this invention so that you can retire one day. Smart, very smart. No wonder you were able to invent a time machine. This guide should help you get on your way, though this is by no means an exhaustive guide.
First, let's set some rules. Number one: you can't just sell the time machine. If that was your goal, why would you be reading this guide? I'm sure there are plenty of buyers out there. Number two: no changing history beyond anything minor. Talking to locals, buying some (small) goods, that sort of thing: totally fine. Preventing Columbus from getting to the Americas? Changing the result of WWII? Really changing the result of WWII? All not allowed. If you want to do one of these things, go for it. But figuring out what would happen to the economy is far too large of a task for this simple guide, so financially you're on your own.
Finally, we need some limits on what you can and can't do with your time machine. I'm going to assume this is one of the first time machines; if it isn't, some government agency has probably stepped in to limit financial gains from the past. And if they haven't, everyone else is probably doing the same thing, resulting in limited profit options for you. That said, this is a relatively untested device. Moreover, the more time jumps you complete, the more chance you have of inadvertently altering history. So to minimize risk, you will go back in time just once, and then return to today's date (which for the sake of this guide is January 1, 2022. You might need to adjust your numbers to account for this fact). You also cannot trust that your wishes will be carried out by anyone you talk to outside of normal situations: if you were around in 1980 and someone asked you to buy some Apple stock for them, wait 30 years, sell it, and buy Tesla, would you do it? If they're a financial advisor, they'd probably have a hard time convincing their boss/regulators that the account is in fact active and they're just waiting for a non-existent componany to go public. And if you're giving your money to someone that is not a financial advisor (like me - nothing in this article should be construed as financial advice, for the past, present, or future), you have a lot more trust in this random person than I do (and keep in mind, even if the "random" person is your ancestor, they won't know that so the chances of them following your wishes are pretty small).
With that out of the way, let's get started!

Before we can even think about when to go to, we run into a problem: how are you going to invest in the past? Your money is probably no good; you won't be able to use a 2020 $20 dollar bill in 1980. And I highly doubt you have any half dismes laying around. So you have a problem: you need cash now whenever you are going to. If your destination is somewhat recent, you may be able to find some old currency and use that, but that's a lot of work. Instead, why not just make money the old-fashioned way? Pull yourself up by your bootstraps and get a job.
What kind of job can we find for you? The obvious answer is to just use time travel. Convince a local sports bettor you're from the future, and give him answers for money. Or show off your fancy future gadgets like a cell phone or an MP3 player. I recommend against this, because it might cause unwanted attention. If you invest in Apple stock in 1980 and then all of a sudden try to sell a huge chunk of stock in 2022, the SEC might investigate. If there's rumors of someone coming from the future and buying stock, they might piece it together and find out you used time travel to invest. And that's almost definitely securities fraud. You're going to need to be discreet; the smaller a trail you leave, the better. Part of that involves not flashing around your future knowledge or technology.
More importantly though, even if you do go this route, you're going to get paid in the local (temporally) currency. You can't ask someone from the 1900's for thousands of dollars for your technology; they won't have that much money. An average yearly wage was $432. It went a lot further, true, but you're still not going to get as much raw currency for your technology as you could in future decades.
So back to the question of your job. I'm going to assume you have next to no skills. True, you did build a time machine. But if we go back to 1900, what can you do? You can't build new technology without changing the future. And unless time machines turn out vastly different than I anticipate, I'm not sure your skills will transfer well to the top-paying jobs of the time, like lapidary work, corset making, or photolithography. You could learn. Or, you could just take a minimum wage job, work a week or two, invest, and be on your way. We can assume you bring some food with you and you sleep in your time machine, so any wages you earn will go directly into your investment. So for the purposes of this guide, we'll assume you work one week at minimum wage (or whatever is comparable before minimum wage becomes well-defined) and then you make your investment. How long you work for doesn't really matter, since everything in this guide will be in terms of return on investment (ROI). The more you work, you more you make, but the change in your investment as a percentage won't change. But if we set it to a fixed unit of time, we get a good comparison of your return on labor. That is, it helps deal with the fact that getting $10 to invest in 2000 is much easier for you than it would be if you go to 1900. As a caveat to this decision, we will not allow fractional investments. If one week of labor only earns you 25% of an Apple stock, you can't invest in Apple, unfortunately.
If any of these rules seem arbitrary, that's because they are. But they keep with the spirit of the original question that prompted this article: what's the best way to invest over a period of 1,000 years, like Fry does in this clip from Futurama:

Fry doesn't get to change his investment or pick it up at an optimal point in history. Instead, he leaves it (in his case, in a bank account) and simply recovers it 1,000 years later. This is the spirit of this guide.
There are lots of problems with this premise, but most of them don't concern us. First of all, we're not going to use Futurama's 2.25% interest rate. Secondly, since we are investing with temporally-local currency, inflation is automatically included for us. One problem we may have is protecting our investment; if we leave our money in a bank account, it will probably be closed. We'll deal with this when we get there, though.