I've done the Everest Base Camp trek. I'm succeptible to altitude sickness, but the trek is done over several weeks so you gain altitude slowly (2-3 weeks), with plenty of time for your body to adjust. The highest we got was the top of Kala Pattar, a nearby peak, about 18,200'. Base camp is about 17,600'.

Kilimanjaro is higher, the true summit is about 19,340', and it's also a non-technical climb. However, guided tours up Kili usually do the ascent in only 4-6 days, which is not enough time for your most non-Sherpa bodies to adjust. If you're considering going, I'd definitely do some hiking around some peaks in the US to gauge how your body reacts to altitude.

The north face of Aconcagua in Argentina can also be done as a non-technical climb, and it summits at ~22,840'. I don't know if it's the highest "walk-up", but it must be close.

Happy hiking!