(edited)
a. For all of these, I would think to expand the category into subcategories. So not just "bank account" but time value of money and NPV, bond details, etc. You learn something when doing this finer description.
b. Maybe it fits under your (2) but radioactive decay is a huge one you did not mention. Applications include carbon (and other) dating, radioisotopes for health use, after the A bomb, or a leak/accident from a power station. If you look at the source, it decays toward zero, if you look at a (stable) product, it approaches a set amount. It is possible to complicate the problems slightly with some algebra by looking at concentration versus activity or different fission products or decay pathways (meaning it is not a 1:1 factor) or with uranium enrichment/depletion (some proportion needed at beginning, end of problem).
c. Analogous to thermal gradient equalization, there are also chemical gradient equalizations or pressure equalizations (through an orifice). [These are not academic problems, a relevant one that I know is buildup of chlorides in a heat exchanger with a pinhole leak to saltwater side.]
d. Capacitor discharge (or charge) in a DC circuit. The former goes to zero (assuming this is the power source), the latter to applied voltage (usually the battery in the circuit). Look up "RC time constant". (There is also "RL time constant" for an AC circuit with an inductor, but high school kids usually grok DC better than AC.)
e. Some chemical reactions (depending on rate order). Usually this is way too fast to be a meaningful problem for tabletop or industrial reactions but if you look at geochemistry or the like it can be meaningful. Note that this is a kinetics problem approaching an equilibrium value.
f. (From googling) compaction of sediments and buildup to terminal velocity (falling in air resistance).
g. Attenuation of a flux through a media (light, X-rays, neutrons). [Note the dimension is no longer time but thickness. So it is a nice difference. but math is still an exponential. Instead of looking at "half lifes" (5.5 years for cobalt 60), you normally talk about "tenth thicknesses" (inch of lead for gamma rays).
h. Decline of an oil well. (Note there are several different models, but the simplest is exponentional, which corresponds to a bounded permeable reservoir.) Look up Arps decline for reference (lots of good explanatory pages on the web so you don't need to, but may, read the iconic journal article itself.) Note also, while it might be too much for an "Algebra 2" student, that the other models (hyperbolic, harmonic, etc.) are quite nice mathematically and you can see how they change into each other via "b factor" (b=1 is harmonic, b=0 is exponential). Might be OK for an honors class or pre-calc class that is doing a lot of function graphing. I had to come up to speed in this field and (having no geology background) found that strong high school functions knowledge helped me to nuke it out. So all that high school stuff can make a difference!
i. Product/market introduction. See "product life cycle" or the work of Christensen from Harvard Business School. The model is concave up exponential at beginning, roughly linear in middle, then concave down to a plateau (in other words a sigmoid curve, with an inflection point). [If the product ends up being displaced, you will have a have a bell curve effect of going down as substitute products come in, and then concave up with the tail of usage. (This is also what graph of the market leader that is losing share looks like.) For a given product, may approximate a Gaussian if the plateau is short in duration or two sigmoids with an approximate level plateau in between.]
j. Another curve that approximates a Guassian is the famous Hubbert "peak oil" bell curve. I would be leery of saying "this is how it works" since of course you can have deviations from shape (Hubbert admitted this, it is a generalization) and even radical departures, like US recent oil production. But if you look at areas like North Sea or Cantarell, model has good fit. And part of life in engineering/science is just to try models and they make work for a while (even a very good fit and theoretical sound fit like buildup of toxins in fish in a stream might change if the hydrology changes for instance.)
k. Of course any normal distribution (arrows hitting a target, observers estimating a value by eye, IQ, etc.) will have a Gaussian distribution. I would probably skip this as it is harder to introduce the more abstract concept of a distribution than to discuss time variance. [But including for completeness.]
Bonus: Not directly responding to your question, but I find that showing people graphing on log-log or semilog scales is helpful in coming to grips and seeing application. You can actually go to the engineering store and get that paper with the tiny green lines and such...you can just feel the science/engineering application come through by being a little physical. (Mathematically it makes no difference, but we are psychological creatures, not computers...while you are there check out the other cool paper like the triangular phase diagrams!)
Lot of common effects are logorithmic. [You can also have the kids graph some functions from above a-h list in linear and semilog paper.] Sonar/sound is one I know well, but maybe little bit of a tangent to bring in decibels. But earthquakes and Richter scale is one that people know of by word and useful to learn it is a log scale. So a 7 earthquake is not marginally worse than 6 quake but ten times worse. And a 3.5 is some wimpy stuff!
