Some common ways that the number of beats will not add up to the time signature:
(1) the final measure of a piece with a pickup measure will often be deficient (add the last measure + the pickup measure and you will get a full measure).
(2) mid-measure repeat signs; usually caused by pickup notes. For instance, a repeat sign after beat 2 of a minuet. One can argue whether the two sides of the repeat sign together constitute a full measure (they are usually numbered as such) but in many computer encodings they will be classified as two measures.
(3) when tuplets are not explicitly marked they will often appear to not fit the time signature; whether this is a case that fits the question can again be debated, but (as in the end of Schumann Carnival where 3/4 suddenly has 4 beats in some measures) they can create problems nonetheless.
(4) Some composers have used "close enough" notes to simplify rhythms which would otherwise use many ties. Donald Byrd has collected several of these at http://homes.soic.indiana.edu/donbyrd/InterestingMusicNotation.html. For instance the Brahms Capriccio in D here: http://homes.soic.indiana.edu/donbyrd/InterestingMusicNotation_files/Brahms_CapriccioDtHalfCtxt.jpg
(5) Depending on how you read the notes, several of the examples in Julian Hook's "How to Perform Impossible Rhythms" may qualify. http://www.mtosmt.org/issues/mto.11.17.4/mto.11.17.4.hook.html
(6) Cadenzas and other such events as noted by others
(7) In "Mensurstrich" and other formats for transcribing early music, it is common to avoid the use of ties as much as possible. So for instance, in 2/4 a four-beat note might be written as a whole note with the next measure left blank rather than as two tied half notes.
(8) Conversely, it was common into the 19th century to write a quarter note tied across the barline to an eighth note as a quarter note with a single dot written on the other side of the bar line (the Bach Gesellschaft edition available on line does this often). Thus the next measure can appear to be short an eighth note.
(9) Feathered beams (https://musescore.org/sites/musescore.org/files/issues/feathered%20beams.jpg e.g.) often result in an indeterminate number of beats in a measure that can only be resolved by context.
(10) Cutaway scores (e.g.: http://www.musica-ferrum.com/shop_files/images/slyorig_1c9c27eb9dc7018746d86c890bd4fc3e.png) will often have measures that do not add up to the right number of beats. The remaining beats are assumed to be rests.
(11) Brian Ferneyhough and other new complexity composers often use non-power-of-two time signatures (2/6) for instance which give an implied tuplet mark, but whose notes do not add up to the "correct" number of beats without it.
There are all sorts of other notations (two-note tremolos, extreme voice layers, etc.) that can be perceived as looking like they don't add up to the right number of beats, but are generally accepted as having the correct number.
Almost all of these (except 1, 2, 3, and 6) are pretty rare and obscure, which is probably why the original website did not list them, but the questioner did ask about less common exceptions. :-)
]
