Inheritances
Could you have a link to the UK's unclaimed estates register or shares?

The Unclaimed Estate Register
The UK government publishes a weekly list of the 7,000+ estates that have remained unclaimed for up to 30 years here. An individual can make a claim for the estate if they can prove descendance from one of the deceased's grandparents.
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Practically this means that an individual who passed in 1991 at the age of 80 may well have had grandparents born in 1851. I'm sure you would agree that most people couldn't name all sixteen or perhaps thirty-two of their ancestors from that period. With complicated family histories and illegitimacy, it may not be immediately obvious as to the identities of those involved.
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It is therefore valuable for you or I to note the names and identities of your own ancestors and compare them to this list. If there seems to be a possibility of a familial link, I am capable of fulfilling this investigation on your behalf, obtaining the necessary documents as required.
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Many successful businesses, styled as 'heir-hunters', work from this list to find individuals that are due an estate. I have found that working up toward this list, although rarer in success, provides a different route that heir hunters are unable to approach from.
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I have been able to connect families to two estates in the last few years.
A Realistic Expectation

You may not think that it's even worth your consideration. I would agree that the chances of finding a familial link are low, but let's quickly look at some maths.
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You would need to prove descendant from an estate's grandparent. If an estate were 20-30 years old and the late owner died at 80-90 years of age, they may well have been born c1910. Their grandparents were likely to have been born in 1850.
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It is not unreasonable that a couple from that era would have had six children in their family that survived and chose to have their own families. Likewise for the next generation. The two generations after that more likely had three families, which might end in you. That's 2x6x6x3x3 = 648. Multiply that by the 7,000 estates and we have a potential of 4.5 million people with a link. That's 1 in 15 people in the UK.
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Obviously, this is an incredibly simplified calculation, but add in illegitimacy, consider loss of contact between family branches, and you can imagine that it may well be worth not dismissing this out-of-hand.
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Now, the government doesn't disclose the contents of the estate. It's extremely unlikely that there's a country mansion falling into disrepair somewhere with your name on it. It may just be a small amount of savings in a bank account. But it is worth checking.

Unclaimed Shares
Unclaimed UK shares and dividends are estimated at £3 billion and cannot be cancelled. They will always be due to the registered holder or their rightful heir.
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As with the estates, it's always worth checking - these shares are unclaimed precisely because people don't realise they are due to them.
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It is usually the job of a solicitor dealing with an estate to make the respective searches for such shares, but personal experience has shown me that, for reasons including previous addresses unknown to them or just plain lack of due diligence, these can be missed (quite evidently).
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This is something that you can do personally for free by contacting the UK Registrars with your enquiries. However, the information you can glean through genealogy research includes finding addresses you may not have been previously aware of for parents and grandparents.