The first one is the Doctor Who story The Time Meddler in which it's revealed, amongst other things, that one of the Monk's previous schemes involved depositing £200 into a London bank in 1968 then nipping forward in time 200 years and collecting a fortune in compound interest. Just how possible might this scheme be? (For the moment we'll ignore the fact that in the Doctor Who universe 2168 is around the end of The Dalek Invasion of Earth which might have caused some disruption to the banking system.)
Now let's start by assuming that either the interest rate remains approximately the same or the Monk was able to get a fixed rate long term account, though that's a big if. Otherwise the scheme could have been derailed by the bank rapidly passing on cuts in interest rates to savers, but not passing on rises so hurriedly. Let's also assume the Monk chose a bank that he knew would be in operation in 200 years time. Whilst there are protection mechanisms for savers if the bank goes under, it's better to use one with the least fuss.
So assuming a fixed rate of 5% (historically the usual interest rate) across the entire period, then exactly 200 years would yield £3,458,516.16 If the interest rate was higher, say 8%, then the outcome rockets to £967,789,916.98 (though if he waited another year he'd be a billionaire). And if he could find a bank insane enough to offer him a fixed interest rate as high as 18% then he'd get £47,580,769,007,364,000.00 Not bad eh?
But there are two fairly major problems. The first is that the Monk would have to somehow be able to claim the account after 200 years' dormancy. He could, I suppose, pose as the descendant of the original depositor, but might run into problems with inheritance tax. Or he could open in the account by posing as a father wishing to create a nest egg for a newly born son, then nip forward in stages and by posing as each successive generation he could transfer the account down. Or he could claim he was an eccentric who wished to help his descendants and make an arrangement for the account to only become active upon the rightful heir claiming it. He could, of course, use an overseas bank in a country that will never have the inheritance tax. The other problem also relates to the tax on interest, though again the Monk could use an overseas bank. A time traveller could be expected to know what they were up to.
And one final problem is inflation. Any major hyperinflation would wipe out the value of the savings. Even normal inflation will reduce it, but so long as the interest rate could keep ahead of that it wouldn't be so much of a problem.
A more interesting case comes in the Red Dwarf episode "Me²". In one scene Holly tells Lister the following:
"It seems when you left Earth three million years ago... you left seventeen pounds, fifty pence in a bank account. Thanks to compound interest you now own ninety-eight percent of all the world's wealth, but since you've hoarded it for three million years nobody's got any money except for you and NorWEB."Now this scene is a wind-up in a comedy series, but the basic principles are still present. This time round we would have to assume that a person who is frozen in stasis (basically suspended animation) for a very long time can resume their financial affairs upon being released, which makes sense, although three million years is stretching it. If we can accept that the human civilisation (or its successor) still exists after three million years and a person's bank account can remain active then what does this yield?
Given that Lister is hardly the most organised of people, I'm going to assume his bank account is just a current account with an anaemic rate of interest - let's say 0.1%. (And I'm also going to assume that the bank calculates the exact interest rather than rounding it to the nearest whole, which is significant for the first few milleniums.) So even after 300 years Lister would have only £23.62 on deposit. After 3000 years it is £350.97. But after 30,000 years it has risen to £184,230,880,712,550.00. And after 3,000,000 years it has risen to £2.9886565148476356806364059915386 x 10 to the power of 1303. That's a very big number.
And this is where the whole thing would get messy. Interest doesn't grow like a tree, it is made by banks taking their depositors' money and investing it, then giving them a return. And current accounts have terms and conditions that allow the depositor to withdraw the lot instantly, so a bank would be vulnerable to a run if it didn't have reasonable funds available. At some point there just wouldn't be any more money that a bank could make and the interest payments would grind to a halt. The anaemic interest rate would, however, mean that the bank wouldn't have that much to make proportionally, but it would still be tricky. They could, I suppose, offer to buy Lister out of the account with a fixed amount and get a court order or special legislation to impose it upon him if he can't be reached, but that might have side effects on consumer confidence. Equally they could, I suppose, take a risk on Lister not returning and use his money to run the economy, but it would be a big risk. But then the entire scene is a joke being played on someone not au fait with all this.
Another jokey one is in The Hitchhiker's Guide to the Galaxy where we learn about how to pay for a meal at the Restaurant at the End of the Universe, located at the end of time which is 576,000,003,579 years away, as counted by Marvin. This is an underestimate according to Wikipedia: Future of an expanding universe will be about 10 to the power of 10 the power of 76 years in the future. My computer's calculator cannot even process that number. For the meal you just deposit one penny into your bank account and then by the end of the universe compound interest will pay for the bill. The computer still can't calculate even 576,000,003,579 years' worth of interest at 0.1% so we'll just have to assume it's a big amount. Again we have to wonder how the banks can manage it, but the whole thing is impossible, but then so is the entire basis of the Restaurant.
Sadly time travel and stasis are not yet available and so we're all mortals on a rather shorter scale. And current interest rates are so low that if you can find a long term account with a fixed rate good enough to grow a fortune for your great, great, great grandchildren then you're very lucky and they will be luckier still.