LOT airlines-round trip ticket?

You won't have problems if you don't fly LOT from Bucharest.
You should look at WOW or Norwegian Air as they sell one way flights cheaper than traditional airlines.

Many people do that and as long as you take the first leg and throw away the second ( the return leg) you should be OK. But why do you want to fly one way? How will you get home? Where does Bucharest fit in? Would a Multi-city/ open jaw flight work for you?

Edit: or is this an internal flight inside a longer trip you have booked RT to and from Europe? In that case the answer is above: fine as long as your flight home from Bucharest is not on LOT.

Technically it can violate the "Rules of Carriage" of the ticket, which is all the fine print you get when you book. Under those rules, the airline can seek compensation for the difference in cost between the ticket you bought and the flight you took. Will they? Very unlikely. If your ticket had another leg that you intended to take, they likely would cancel that leg. If you fly with them shortly after skipping a flight, they may have you flagged and cause you issues at check-in, but in general they know they cannot do much. Only partly related, many did this for years on Eurostar, buy a cheap "return same day" ticket and throw the return ticket away once in Paris or London.

There used to be an app that searched flight schedules and found deals to out of the way places that flew through the destination you wanted to go to, with the recommendation to throw away the onward tickets, the airlines had a fit for awhile, things died down, not sure if the app folded or the airlines figured they had no recourse.

I wouldn't do this. Surely you can find a ticket that works both ways. Many people are outraged that airlines overbook seats due to some probability of "no shows"...but no one seems outraged when passengers themselves violate the terms of the contract knowingly. Just because you can doesn't mean you should...that's my take. That seat you plan on not using (with the cheap fare) would be better off going to someone else who really needs it. It seems like a small deal but if everyone did it, it would destroy the pricing market of airline fares, and hurt everyone in the process because airlines would have to re-jigger their pricing assumptions altogether. I say "keep it honest" despite temptation.

The flight I am talking about is from Krakow to Bucharest. I am flying Delta to Berlin, and then Delta from Bucharest to home. The other airline that I am considering is Aegean air which has a 4+ hr layover in Athens. This would be a one way flight and is a reasonable price. It would cost more than the LOT round trip because I would have to pay to check my carry on as I could only have 1 bag on the plane, and I will have a personal bag, too. I thought that if I booked the LOT tickets, I could call a couple of days in advance and cancel the return flight so that the airline would have a heads up. I have checked a lot of airlines, and these two have the best rates.

This is something I've never understood. Honestly, people accidentally miss flights all the time and the airlines typically tell them "tough luck". But when one does so on purpose, it's suddenly some sort of moral failing? Either way, that seat is typically filled with another paying passenger. Also, not so sure that this is the best week to "call out" passengers simply for contemplating violating the spirit of an airline's ticketing agreement.

To answer your specific question, one way to ensure your flight home wouldn't be affected (assuming the airline doesn't employ mind-readers), would be to schedule the return flight on that cheap RT ticket so that it departs AFTER you're already home. Again, not advocating anything one way or the other, but if one were to do that, AND they then contacted the airline of the unused segment days ahead of departure to give them plenty of notice that it won't be possible to make that return flight, (meaning the airline could re-sell that seat and collect double the profit!) one could argue that their bases would be covered.

Hopefully, my thought-crimes on creatively navigating the byzantine structure of passenger "agreements" don't get me digitally dragged down the aisle by fellow posters for "re-accommodation"... ; )

Maybe you should have a look at www.skyscanner.com which covers European budget airlines, some of which price on the basis of one-way flights rather than round trips. I see one-ways in September that are cheaper than the round-trip LOT fares, plus you would save all this stress about dodgy tactics. The drawback is some of these itineraries require a long layover; I don't see non-stops.

She is talking about a short internal flight from Krakow to Bucharest that is totally independent of her roundtrip flight from and back to the US. there is no way that failing to show up for the second flight on LOT will impact her longhaul flight. Not is it an immoral act to book a return ticket and only use it one way. No one will be harmed. Just don't choose a seat on the second flight, and do set that second flight on LOT far enough out that you can cancel it a week or so ahead of the date.