The answer to this question is probably impossible to determine; self-reporting measures such as the U.S. Census generally tend to greatly underestimate actual multiracial population. For example, genetic testing indicates that 20% of African Americans are at least a quarter European, and that the average African American has around 20% European ancestry. Similar studies reveal that ~30% of white Americans have detectable African ancestry (relatedly,
this controversial paper published in 1958 analyzed population trends in past census records to estimate that 21% of white Americans living at the time had black ancestry). Also, most Hispanic/Latino Americans have some degree of both European and Amerindian ancestry (and maybe African or some Asian blood as well), even though many Hispanics with mixed heritage still self-identify as white on the Census. To really answer this question you'd have to determine what proportion of mixed racial ancestry would be required for one to be "multiracial", then use studies and census results to estimate an answer.
My guess though would be Maverick County, Texas (95.7% Hispanic, equal to a few other Texas border counties, but much of the non-Hispanic population in Maverick Co. is the Kickapoo Indian Tribe which only requires 1/4 tribal blood for membership).
However, this all ignores the fact that race is really just a social construct and is thus entirely arbitrary. "Mestizo" might as well be a race, given the hundreds of years it has had to develop, and similarly "African American" should probably be considered a discrete racial group as well. With race being a completely subjective category, what is considered "multiracial" is entirely subjective as well.
