So, a few disclaimers before I get into this list.
This is my personal favorites list, in no way do I claim that these are "best" moments in sports, although a lot of them I do consider top 8 best sports moments since 2013, and some of you might agree.
Another disclaimer: I don't really follow baseball or hockey at all, and basketball I've only recently gotten into, my main sports are American football and association football/soccer, so the list will heavily skew in that direction.
Anyway, here's the list. It's not ranked, so don't read too much into the order.
1. September 2015-Robert Lewandowski Scores 5 Goals in 9 Minutes vs Wolfsburg
This will never make sense to me with how it happened. The fact that Lewandowski came off the bench as a substitute and then tore Wolfburg apart for 5 goals in such a close stretch stands as one of the greatest 9 minute performances of any striker ever as well as one of the worst defensive collapses by a team ever (but perhaps not the worst, see below when I get to Brazil vs Germany). A few of Lewandowski's goals were simple tap ins, but goals #3 and #5 were absolutely sublime, the work of a consummate striker with a gorgeous scoring touch.
2. June 2016-LeBron James Blocks Iguodala to Help the Cavs Win the NBA Finals Vs Golden State
LeBron James will probably be remembered as the greatest basketball player ever one day (sorry, MJ), with many today who already consider him as such. Watching a clip like this, it's hard not to agree. That block was absolutely venomous, and if James hadn't sprinted like a madman to close Iguodala down, the Warriors probably win the game and the series.
3. October 2017-JT Barrett Forgets How to Throw Incomplete Passes Vs Penn State
Two things: 1st, this moment was admittedly tarnished (but only a little) by Barrett's less than stellar performance the next week vs Iowa, and 2nd, I don't know if it would have been as glorious if anyone but Gus Johnson was commentating for it. This will go down in history as perhaps the single greatest 4th quarter performance by any Buckeyes player at any position ever, but he couldn't have done it without lights-out play from his O-line and receivers.
4. July 2014-Tim Howard Breaks World Cup Save Record with 16 Saves Vs Belgium
This game was a tough one for my family. I'm a dual-national with a Belgian passport in addition to my American one, and my mom was completely rooting for the Red Devils against the USMNT. IN the end, the talent gap was too great and Belgium beat the US in extra time, but Tim Howard stood tall and had a game to remember, breaking the World Cup record for saves in a single match.
5. December 2015-Aaron Rodgers' Hail Mary to Beat the Lions
https://www.youtube.com/watch?v=r0vVqStvh_8 (NFL doesn't let you embed, apparently, go figure)
The pass is incredible, the fact that Rodgers was able to sling it that far with any degree of accuracy and have it caught counts as a kind of sporting miracle.
6. December 2014-Ezekiel Elliott Goes 85 Yards Through the Heart of the South
The run. The superb blocking. The Disappointed-Saban-Face. The Bama girls with their floppy flags. The eternal rhetorical question: "Can They Catch Him? No They Can't!" For a Buckeye fan, this moment was pure elation, euphoria. This was the moment I realized we didn't just have a shot at beating Bama, but that we were certain to win, that we could beat anyone, that we were going to win it all.
7. July 2014-Germany Destroys Brazil at Home
I watched this game at a family party in Belgium. No one else was watching the tv--at this point Belgium was out. After Germany scored the first goal, everyone abandoned the TV set, assuming that Germany would play a conservative game and grind out a boring 1-0 win. Boy, were they wrong. But no one would have ever imagined how bad it would get for Germany. If ever there was a time to use the Anchorman "that escalated quickly" meme, it would be for this match. In the space of fifteen minutes, Brazil went from down 1-0 to down 6-0. By the end of the match, the proud, soccer-loving Brazilian fans were streaming out of their home stadium, and those that remained booed their own players and cheered the absolutely ruthless Germans, who to their credit stopped celebrating their goals after it was clear they'd won the game.
8. December 2014 (2014 was an amazing year for sports for me)- Joey Bosa Shrugs After Scooping and Scoring on Wisconsin
What a game that was. The Buckeyes' 59-0 rout of Wisconsin to set the stage for their national title campaign will always be the high water mark for Ohio State teams in terms of what constitutes a dominant performance. The moment Joey Bosa scored a touchdown off a Melvin Gordon fumble was the moment of no-return for Wisconsin, the moment it became absolutely clear that the Buckeyes had conquered them.
Feel free to post your own favorites below!