Ashley Tisdale -- Fan Tweets Actress 18,000 Times ... Cops Investigating

Ashley Tisdale's legitimately freaked out ... after an obsessed fan sent more than 18,000 tweets to the actress and then showed up to her home, TMZ has learned.

Law enforcement tells us ... the LAPD has launched an investigation after a person in Ashley's camp reported that a super-fan had tweeted her 18,888 times, dating back to November, 2012.

The tweets are pretty creepy -- the person (believed to be a man) seems to be delusional, writing as if the two have some sort of ongoing relationship. For example:

"My body is yours & even as friends I f***in love how I make u feel like mom maybe we both got issues cuz nobody's perfect."

"I'll never lose u baby even in 20 years til infinity you'll always b my best friend."

"Let it all out on me u can have my heart put a lock on it throw away the key & my soul is yours to keep."

Even scarier ... we're told the suspected super tweeter recently showed up to Ashley's house ... but we don't know if he ever made contact.

We're told cops have identified a suspect -- and are looking to talk to the person to determine whether he poses a serious risk to Ashley.