I love Calculus. Give me an integral. Make it as large as you like. I’ll solve it or die trying. I like using Calculus to solve problems.
Which field involves the most Calculus? This can be any sort of field whatsoever.
Actuary?
Teaching calculus. 
There are real-world job applications of calculus are few-to-none. Any problem with a closed-form solution is trivial and will be found in a book of canned equations. Real problems are solved with numerical analysis.
Solving integrals is a neat puzzle, figuring out what integrals need to be solved is where the work is.
As an economist, I’d like to suggest that the answer is physics.
Physics, or anything that needs to measure or interface with the real world. In my field (CS) the subfields that use the most calculus are machine learning, machine vision, robotics etc.
I can’t speak for accuracy, but this website is a specific list of jobs that use (not just require as a course) calculus.
http://www.xpmath.com/careers/topicsresult.php?subjectID=5&topicID=1
If you’re not a researcher, it’s almost certain that you won’t be doing math on the job. You’ll be figuring out how to enter your problems into tools that other people have developed and how to interpret the output.
General relativity. Every tensor equation is implicitly one or more integrals.
Hmmm… I ran across a PLC program in a smelter that linearized the square root function. Within a certain range, the answer was accurate enough - within 10% - that it was close enough. Beyond a certain range, the controller was acting a bit off; however, it still did enough of a job that the process ran. It sure sped up the processor, which was about as fast as a slow VIC-20.
Many process control units use PID feedback - integral and differential - the further off course, the harder you correct, and the longer off course - integral of error over time - the harder you correct. Of course, these are canned functions, so you don’t actually need calculus yourself.
Over -infinity to infinity, integrate [1/(x[sup]2[/sup]+1)[sup]2[/sup]] dx.
Eggs, spam, bacon, calculus and spam - there’s not very much calculus in that.
oh wait, you want the most calculus…
I tutored a guy who was doing a master’s in explosives engineering. I don’t know how much of it is used in practice, but the math used to discribe the motion of shock waves as they transitioned various materials was pretty wicked.
Example: Funeral Director
Yeah, well. As I said, I cant vouch for complete accuracy. The engineering etc looked good
I am a computer programmer and the highest math I ever used at work was basic trig. As far as engineers are concerned, it seems they deal more in rules of thumb than anything, but the ones I know aren’t designing major products either.
I’ve solved a few applied problems using closed form calculus, like wave propagation across a symmetric plate or member, and some simulation code for bending modes. I’ve had more use for differential equations, particularly in time dependent reactions as in interior ballistics. And even solutions that are solved by numerical analysis still use calculus, often fairly complex, multi-variable, and variational calculus.
I would say that the applied field that directly uses the most calculus is analog signal processing and filtering. You just can’t get into a good discussion about signals without writing half a dozen integrals on the board. Any kind of physics simulation (finite element method, computational fluid mechanics, electromagnetic field interactions) uses integral calculus and/or differential equations, but for the end user these are hidden in the code; only the developer or a power user really has to think about the math too much.
Stranger
I believe the correct question was: over -infinity to infinity; integrate x[sup]x[/sup].
ETA: Or does that violate the rule against wishing death on other posters?
Computer graphics, in particular “production” rendering (e.g. film CG as oppossed to “real time” graphics such as games) uses ALOT of calculus.
My previous job involved “real time” graphics, and while it was very maths intensive, there wasn’t much calculus invovled (or at least there didn’t have to be, there was alot of calculus invovled in deriving the various algorithms, but you could usually skip through the calculus and look at the code).
I now do more “production” graphics and that’s not an option, you actually need to understand all the d-whatevers and funny S symbols 
Though in terms of the MOST calculus I’d second Stranger On A Train’s suggestion of numerical simulation.