Well this applies to many, many species of insects. While a few different sex determination systems are at use in insects, including xy (fruitflies), xo (grasshoppers), and zygotism (ants), all of them other than xy involve differing chromosome numbers between males and females. xo is the system you discuss in your question, where females have an extra x chromosome and males don't, with no equivalent to the y chromosome present. In this case, females always have one more chromosome than males. In zygotism, the sex determination system seen in ants, bees, and most wasps, males are haploid (possessing only one copy of each chromosome) and females are diploid (possessing 2). This means that females have twice as many chromosomes as males. Interestingly, it is this odd form of sex determination that appears to have driven the evolution of these species into eusocial insects. If two sister bees share the same mother and father, they each got half of their mother's DNA and all of their fathers DNA, since he is haploid. That means that they share 50% of DNA in common with their mother and any children they might produce but 75% in common with eachother. Due to this genetic eccentricity, a bee gets more of her DNA out there by protecting and helping her mother push out more sisters than she would by having children of her own.
Outside of the insect world, we have other mechanisms for inducing chromosomal differences that are not related to sex determination. You mentioned trisomy but this is really just the tip of the iceberg. Polyploidy, mutations resulting in more than 2 copies of one or more chromosomes, and anuploidy, the deletion of one or more chromosomes, are very rare in humans, with trisomy 21 being the only really publicly well known example, but common in some other species. Ferns and other simple plants regularly duplicate all of their chromosomes, resulting in triploid and quadroploid offspring that are still viable and able to produce children of their own. Mice are also very susceptible to this type of mutation, to the point that there are currently 97 separate populations of the common house mouse characterized by various chromosomal eccentricities. Here is a rather interesting study on this tendency in the common house mouse.
Chromosomal variation in the house mouse