Yearly Jar: Each payday it takes £x where X = Sum of sub-targets / pay days. Targets may or may not have a yearly date (e.g. Xmas, Clothes)
Bonus if it tells you how much you have for each sub-target or lets you re-allocate.
Edit 1
Blind Jar/Piggy Bank: Deposits only until broken open. Saved amount may or may not be visible. Inspired by Schrödinger jar below, but without date limitation.
Assign a DD/SO to a Jar Jar
Allow one or more DD/SO’s to be assigned to one or more pots so that bills can be paid out of it. If the money isn’t present in the jar it offers a settting where it can drop back to the main account
Overkill Jar - save all disposable income.
Optimal Jar - save and invest to maximise my utility.
World Traveller Pot before you go on holiday you move a set amount of your hard earned money to your World Traveller Pot.
By land sea or air you arrive in your foreign destination and you get your nice notification from Monzo welcoming you to your destination 
All transactions in that foreign currency are now taken out of your world traveller Pot. You enjoy the sites, you enjoy the food and you enjoy the drink without effort or worry.
Then as with all good things your travels comes to an end and by land sea or air you are back in the uk sad and wishing you were back in the sun. But Monzo keeps you smiling and all payments revert back to using your Monzo current account … effortless travel money!
Your pot could also notify you if you are getting low on funds to allow you to top up or could if funds were available move from the current account to the traveller Pot to keep travel separate.
I like the idea of the Guilt Pot mentioned earlier, every time I spend at JustEat or McDonalds, KFC etc then whatever I spend is also added to my Guilt Pot. Once I have lost a few Lbs my £££s will be released back to me for new clothes
This. Definitely this.
Pyramid Jar (may need a better name…) - Basically the same as the 1p Saving Challenge, but reordered to ramp up from January to July and back down again from July to December, to spread/mitigate the effects of the holidays.
Pot of Gold 
Just like the one at the end of a 
Randomly, someone gets this pot with free money (17th March would be a good date to do this 
)
Monthly Outgoings: At the start of the month, when I get paid over £x,xxx amount, move £yyy to a monthly outgoings pot.
This outgoings pot would be used for regular monthly direct debits (phone, magazine subscriptions, web hosting, utility bills)
Monzo should detect recurring automatic payments and assign to this pot.
Definitely this, but why wait to settle daily? Instant Access Savings Keep any positive balance in an interest bearing pot, with funds automatically withdrawn as and when needed. Maximise interest without worrying about ever accidentally triggering an overdraft.
Swear Jar every time you utter an expletive you transfer £1 to this jar. Could use IFTTT to create a swear button / icon to do this. Even better - the facility for your colleagues to trigger the transfer (still to your own pot obviously).
Lunch money every time you are paid your salary X amount is transferred to the pot which is then released slowly back into your account at a rate of Y per day (at whatever hour you set as lunchtime!)
Edit:
Flexi Jar Talking of IFTTT earlier got me thinking why not allow some user selectable options so they can create their own tailor made jar themselves.
Pay my expenses Pot Sweep incoming expenses payments from [my company] into my Expenses Pot and pay my [Amex] Direct Debit from my Expenses pot
Pay my [bill] Pot I will add £x to my [bill name] Pot and my [company Direct Debit] will be debited from this pot
Beat my target Pot At the end of the month, sweep the amount I’ve not spent under my [category] target into my [Category] savings Pot
Unjarred - liberates you from the jarring sensation induced unpredictable savings.
TL;DR: One big pot with an associated list of items (TV, holiday, house deposit, …) that you want to buy. The user inputs the items and their costs, and the app calculates the date at which they will be able to purchase each. The list is re-orderable, allowing the user to quickly and fluidly create future scenarios. The app knows the user’s effective savings rate which, when combined with the item costs and item order, is used to calculate the dates. Transfers happen into the account as infrequently as possible - ideally once per income period (typically a month), but only at the end of the month once the money is definitely saved.
TL;TL: I wanted to provide more detail on my thinking, apologies for length.
I think that it helps to consider savings in their essence - discretised deferred spending. One pot per item, the current model, is overly discretised and makes it hard for people to know how much to put in each pot, and makes it hard to fluidly reprioritise the items. The pots feature should allow the user to view different predicted futures, so that they can decide on the one that they want to actualise. To achieve this, I suggest one large pot in combination with a list of items and prices of those items. Then, in combination with the predictable savings rate (see below) you can allow the user to reorder the items in the list and the app can update the date that they will be able to buy each item.
How can we get predictability in the savings rate? I suggest minimising the number of transfers both in and out of savings. Ideally there would be one transfer in per month, and only purchases outgoing. Many of the ideas here are variations on "if <arbitrary condition>: <transfer money to pot>". I think that this approach is flawed because it doesn’t take into account what it’s actually possible for the person to save. They’ll either save too little (and potentially end up with needless discretionary spending) or save too much (and have to tap savings to meet expenses - reducing predictability). However, people didn’t suggest these strategies for no reason, and I think the unifying element is the laudable “get the money out of my sight ASAP!”. If you don’t have clear edges to your expenses - and it all appears in one big pile in your account all month - then you get numb to your day-to-day discretionary expenditure.
Thus, to minimise expenditure, maximise savings, and provide predictability over savings, I suggest the following 4-pot “buffer” strategy:
Discretionary pot
Bills pot
Buffer pot
Savings pot
In the current strategy Discretionary, Bills, and Buffer are all in one big pot (your main account), and bills & large expenses will be hidden from the spending target by flagging them individually. I think it would be a lot easier to plan if these different kinds of expenses were segregated more clearly in the app.
The key is to provide predictability, and hard (psychological) edges around the savings. Once it’s in there it shouldn’t be used for arbitrary discretionary spending, or to meet the expense of an outsized bill. Once a few months of data have been logged then the app will be able to show predicted savings into the future and thus be able to assign dates to each of the items listed in the pot and allow the user to reorder items and create possible future scenarios.
Credit Card Jar: Use Open Banking APIs to monitor my credit card spending and ring-fence enough money to cover the bill at the end of the month
Locked coin jar - putting a sealed until date on the coin jar so that you can’t transfer out of it until after that date
Naughty Jar - If you succumb to buying fast food you are fined (such as KFC, McDonalds, KFC, Greggs, KFC, Burger King, KFC etc). The fine to be placed into a jar (and preferably locked for a period of time, or difficult to retrieve). The funds being locked could be an extra incentive in resisting the temptation. The jar would work in 2 ways. It would either save you a lot of money from not buying all those fast food meals or more likely if you are anything like me the fines would rack up.
Regular bill jar - Any direct debits or standing orders would be taken from this jar. When you are paid the estimated value of the expected direct debits would automatically transfer into the jar. To start with it may not be accurate however over time Monzo would better learn how much is required and be more accurate. Any surplus at the end of the month could be transferred to the coin jar.
Schrodingers Jar. You can transfer money in but the balance always stays zero until a specified date.
Bills coin jars Very similar to the idea my gran had when I was a kid - saving money in a jam jar for different bills that were coming up. Could be taken away from salary when credited to the account - so the balance in the account is what you’ve actually got spare to spend after bills etc.