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If I've got a flight booked to a country that requires an e-visa for entry (say Brazil, for example), and my visa hasn't been processed yet, but will most likely be processed within 24 hours of my flight's arrival, is it safe to board my flight, and then just wait in the transit area of the airport until my visa is approved? Or could I get in trouble for even arriving at the airport without a visa?

Would it be helpful to buy a cheap onward ticket for the next day, just in case I get asked any questions about why I'm waiting around in the airport?

Note: I'm making a few assumptions like 1) the airline doesn't check if I have a visa and I'm able to board the flight 2) the visa will be approved 3) I am flying with only carry-on bags 4) I already have an international SIM in my phone 5) if the visa isn't approved for any reason, I could just buy an onward flight to somewhere else.

You won't get in trouble for arriving at the airport without a visa. Instead, the airline you're flying won't let you on the plane to depart in the first place. They can check the status of electronic visas at the time you check in. They do this because the airline is responsible for transporting you back out of the country if you are refused entry. If you don't have a valid visa, and you need one, you will be refused.

As Greg Hewgill notes, the airline will not allow you to board. The airline will generally use a database of visa requirements (e.g. Timatic) and will determine what documents you require based on the passport you present. So if you present a passport that requires a visa to enter Brazil, they will know you require a visa, regardless of what country you're flying from. This is generally built into the airline's IT system and is not simply something they may forget to do even if you're flying from a country where many people don't require visas.

Buying an additional refundable flight out of Brazil wouldn't work either, because Timatic will tell them that you can only transit without a visa if you have "a connecting flight booked on the same ticket in transit." As stated on a Brazilian government site, you need a transit visa if you have separate airline tickets to change planes in Brazil. The airline will enforce this rule and deny boarding as well.

In addition, not all countries and airports allow you to remain in the transit area overnight, so even if you make it there, you could have a problem if you stick around for long enough.

Judging from my personal experience (primarily with JFK and MIA), your tickets will be checked by the airport staff/security regardless of whether you're going to clear the customs or proceed to the transit area. If there are no transit passengers on board, there may not even be an option to get there as the corresponding door would be locked.

While this is impossible to say whether or not you would be "lucky" with the visa check upon departure and the ticket check upon arrival, generally such a strategy sounds like a risky undertaking.

Here are some more things you should be aware of:

1) the airline doesn't check if I have a visa and I'm able to board the flight

Greg and Zach both mentioned correctly that airlines do check your eligibility to be landed at the destination; as they are fined and are responsible for repatriation if you are inadmissible.

2) the visa will be approved

This a dangerous assumption to make, and as per #1, a moot point.

3) I am flying with only carry-on bags

This really doesn't have an impact on your waiting in the transit area. At some airports, you can freely move between the transit and the landing areas. In others, transit areas are limited, restricted, or may be closed entirely.

4) I already have an international SIM in my phone

Not really relevant to your original question, not sure why this is even a concern.

5) if the visa isn't approved for any reason, I could just buy an onward flight to somewhere else.

You are assuming that you will find a sales desk in the transit area for the airline you want to take. This is risky. In my experience, transfer desks in transit zones don't sell new tickets (they can only assist with existing reservations and upgrades / downgrades). I have had many people whose flights were cancelled try to buy tickets on another airline but the agents were unable.

If an onward ticket is required for your visa, this will also be checked by the airline before you are boarded.

Gone are the days that you could buy a ticket from the gate agent - these days you need a ticket just to get past security.

Would it be helpful to buy a cheap onward ticket for the next day, just in case I get asked any questions about why I'm waiting around in the airport?

At some airports you may be questioned if you look like you don't belong - as to what you are doing in the transit area. A ticket may not help you, since you would need a boarding pass to really justify your presence in the secure area.

In some airports, there is a limit on how long you can stay in the transit area (before you must be landed or depart).