Thanks to his brilliant intellect, Sheldon was one of the best theoretical physicists at Caltech in The Big Bang Theory. He enjoyed his work so much that most times, he refused to go on vacation until he was forced to do so. Although his dedication to his work was commendable, it was not without its issues.
While Sheldon rarely ever missed a single calculation, he wasn't the sharpest at knowing where to draw the line when it came to inappropriate behavior at the university. If it wasn't something he said, it was something he did, either by himself or with the rest of his crew. He got away with a lot of it, even when some of it called for immediate termination of his employment, or at least, a suspension.
Foamy Revenge
"The Vengeance Formulation" had Sheldon lined up for an interview on Science Friday, a popular program on the radio. During the interview, Barry fed a pipe carrying Helium into Sheldon's office, which caused his voice to get higher and almost cartoon-like. Embarrassed and enraged, Sheldon decided to get revenge on Barry with a prank of his own.
He set up some chemical reagents in the ceiling of Barry's lab, which would be triggered by motion detectors once Barry went inside and become foam, which would drop from the ceiling. Unfortunately, Barry wasn't alone and the foam landed on the president and members of the board too. To add to that, Sheldon recorded an audio clip claiming responsibility for the prank, so there was no way he should've gotten out of that one.
Buying Chemicals Illegally
During one of the few collaborations between Sheldon and Leonard in "The Helium Insufficiency," they found out other scientists were trying to prove a theory they'd come up with. Worried that they'd beat them to the punch, they decided to do experiments to prove the theory before they did. Unfortunately, there was a limited supply of one of the main reagents they needed, so they resorted to buying it illegally.
When they took the Helium back to the university, they noticed there was a label torn off the container that indicated it might have been stolen from a federal facility or another university. Thereafter, a back-and-forth ensued as they grappled with whether or not they should use it. Not only was this illegal enough to land them in jail, but it could also have cost them their jobs as well.
An Egg Salad
In season 6, Sheldon hired an assistant named Alex. When he realized that Alex was flirting with Leonard, he decided to give her "a talking to" so she could refocus her attention on him and his needs. The talk went badly, to say the least. Using vague similes and odd comparisons, Sheldon compared Alex to an egg salad sandwich and told her she was a slave to her urges.
To dissuade her from falling victim to those "urges," he also showed her a book with graphic depictions of venereal diseases, which made Alex, understandably, very uncomfortable. In the real world, this would certainly qualify as sexual harassment and would be grounds for dismissal.
"Even You Are A Slave"
When Alex reported Sheldon to the Human Resources Department for all the inappropriate things he'd said about her and her body, he was called in to speak to Mrs. Davis. Sadly, no matter how much she tried to explain to him why what he'd said was wrong, Sheldon simply did not understand and believed he was right.
He even went so far as to say that Alex may have misunderstood by thinking that he was singling her out, hence the infamous line, "even you, you're a slave," to Mrs. Davis, referring to how she was also prone to giving in to biological urges.
Roots
In season 6, episode 12, "The Tenure Turbulence," one of the professors at Caltech died, leaving a tenured position up for grabs. Sheldon, Leonard, Raj, and Barry all had their eye on the position and engaged in a series of hilarious and somewhat inappropriate antics to gain favor with the tenure committee. As the Human Resources representative, Mrs. Davis was one of these, which was unfortunate for her.
Leonard joined her at the gym (uninvited), pretending to be a frequent visitor himself, even though he was clearly out of shape. Barry sent her a banana loaf and watched her kids, Raj made a video with terrible photoshop images of his childhood, and Sheldon, well, Sheldon bought her a DVD of Roots, the TV series.
The French Maid Impersonation
Sheldon mocking Howard's intellect was nothing new and he usually got away with it, but on one occasion, his mockery almost cost him some face-to-face time with Stephen Hawking. Howard was selected to help Stephen Hawking with his wheelchair in season 5's "The Hawking Excitation." He planned to introduce Sheldon to him until Sheldon said something condescending to him.
After that, Sheldon tried to apologize, so that Howard would change his mind. In place of the apology, Howard made Sheldon do a series of tasks, akin to Hercules' trials. That included making Sheldon parade through the university cafeteria in a French maid's uniform he'd bought for Bernadette. The violation of the dress code here at the university was definitely breaking the rules.
M.O.N.T.E.
It was never always "all work and no play" at the university when it came to Sheldon and his friends. In fact, in one episode, the crew joined the Southern California Robot Fighting League Round-Robin Invitational and built a robot they named M.O.N.T.E. They took the robot to the university and did a trial round with Barry's robot, the Kripke Crippler, which led to a loud, disruptive chase through the university hall and the destruction of university property.
In addition to all the things damaged in the process of fighting these robots, they used university resources to build something purely for recreational purposes, which was also a waste of money and against the rules.
Desk Wars
In one of his few kind gestures, Sheldon let Raj work for him so he could retain his work visa and not be deported. That meant they had to share Sheldon's tiny office space. Raj asked if he could get a desk in the office, and after refusing for a while, Sheldon eventually agreed. In retaliation, Raj got a huge desk, described by Sheldon as a Brobdingnagian monstrosity. It took up half the room, making it almost impossible to get in, let alone move around.
They bickered over the desk and it escalated to Raj playing music loudly, getting a pet bird, a pellet gun war, and burning aroma candles. Sheldon in return, created a flammable formula of hydrogen sulfide and ammonia gas to create an unpleasant smell, but it ended up exploding. For the destruction of university property and what surely must have disturbed other university workers and attenders, Sheldon should have been fired.
Insulting University Officials
It's no secret that Sheldon looked down on almost all his peers and didn't keep that a secret. His condescension extended to Doctor Gablehauser, introduced as the head of the Physics department in "The Luminous Fish Effect." Upon meeting him, Sheldon introduced himself, stating that he was an actual, real scientist, implying that Gablehauser was not.
Unsurprisingly, he was sent home right after. This was one of the few, rare occasions when Sheldon received a punishment that actually matched the crime. After his mother forced him to apologize, he got his job back, which might have been too easy.
Every Time He Threw A Tantrum
Throwing a tantrum was as natural as breathing for Sheldon. Whether it was over the university not wanting him to change fields of study or refusing to meet with donors, Sheldon always had something to fuss about.
The university put up with it, probably because he was one of their best scientists and they knew it was in their best interest to keep him around and happy. And boy, did he take advantage of that. His tantrums often led to him saying or doing something inappropriate, which he never got punished for.