Biggest Cross-Browser Issue? - Internet Explorer!
IE is solely responsible for "holding back the web" - us developers can't create amazing sites using HTML5, or SVG, or XFORMS, or CANVAS... not because of Firefox,Safari or Chrome, but because 2/3s of the Internet is still stuck on IE. Not to mention that ~20% of the web is still stuck on IE6! IE8 is the first version of IE to at least try to be standards compatible (2001's standards that is), which means it will be at least 2018 before we can finally start dropping all support for IE7.
Otherwise name a DOM method that IE fully supports... the fact that this is a hard question to answer is my biggest CrossBrowser issue.
getElementById() - badly broken
getElementsByName() - buggy
getElementsByTagName() - buggy
getAttribute() - buggy
setAttribute() - majorly broken
createElement() - buggy
appendChild() - buggy
even things IE invented are messed up...
innerHTML (getting) - returns the worst markup possible
innerHTML (setting) - doesn't work on the elements you'd want to dump a bunch of data into (e.g. Tables and Selects)
