Modern load balancers have very high throughput capabilities (gigabit). So unless you're running a huuuuuuuuuuge site (e.g. google), adding bandwidth is not why you'll need a new pair of load balancers, especially since most large sites offload much of their bandwidth to CDNs (Content Delivery Networks) like Akamai. If you're pumping a gigabit of un-CDN-able data through your site and don't already have a global load-balancing strategy, you've got bigger problems than cache affinity. :-)
Instead of bandwidth limits, sites tend to add additional LB pairs for geo-distribution of servers at separate data centers to ensure users spread across the world can talk to a server closest to them.
For that latter scenario, load balancer companies offer geo-location solutions, which (at least until a few years ago which was when I was following this stuff) were based on custom DNS implementations which looked at client IPs and resolved to the load balancer pairs Virtual IP address which is "closest" (in network topology or performance) to the client. These days, CDNs like Akamai also offer global load balancing services (e.g. http://www.akamai.com/html/technology/products/gtm.html). Amazon's EC2 hosting also supports this kind of feature for sites hosted there (see http://aws.amazon.com/elasticloadbalancing/).
Since users tend not to move across continents in the course of a single session, you automatically get affinity (aka "stickiness") with geographic load balancing, assuming your pairs are located in separate data centers.
Keep in mind that geo-location is really hard since you also have to geo-locate your data to ensure your back-end cross-data-center network doesn't get swamped.
I suspect that F5 and other vendors also offer single-datacenter solutions which achieve the same ends, if you're really concerned about the single point of failure of network infrastructure (routers, etc.) inside your datacenter. But router and switch vendors have high-availability solutions which may be more appropriate to address that issue.
Net-net, if I were you I wouldn't worry about multiple pairs of load balancers. Get one pair and, unless you have a lot of money and engineering time to burn, partner with a hoster who's good at keeping their data center network up and running.
That said, if cache affinity is such a big deal for your app that you're thinking about shelling out big $$$ for multiple pairs of load balancers, it may be worth considering some app architecture changes (like using an external caching cluster). Solutions like memcached (for linux) are designed for this scenario. Microsoft also has one coming called "Velocity".
Anyway, hope this is useful info-- it's admittedly been a while since I've been deeply involved in this space (I was part of the team which designed an application load balancing product for a large software vendor) so you might want to double-check my assumptions above with facts you can pull off the web from F5 and other LB vendors.