Graduate Admissions - applying math phd without outside experience I am currently a senior in a somewhat big state school.

I am going to apply phd next semester for 2018 fall, looking forward to attend top 25.

My gpa is 3.86/4.00.
Havent took GREs, but i think I will do them just fine. 

But heres the thing: Whenever I see posts from one who made it into top 25, most of them had at least one REU and had taken a few grad courses.
 None of them I have done, nor will I because next semester will be my last semester. 
I am really worried that this lack of experience would make me less competitive for top 25. 
How much would this lack of experience (REU or grad courses) make me less competitive for application? 

One more question, actually I am going to take seminar class next semester. After first few lectures, each one of us will prepare for presentation about assigned textbook chapters. Does this class count as a research experience? 
I know this question could be answered most clearly by my prof. I just want to listen your opinions.
Thanks for your comments in advance.
REUs probably don't matter all that much. Presenting textbook chapters will not count as research experience, although depending on the material it might count as evidence that you can learn advanced material on your own. So definitely get a letter from that prof if you do a good job.

From what little you've said, you're probably only semi-competitive for a top 25 school, and probably not competitive at all for a top ten school even with stellar GREs. But there are a lot of factors in play. For example, not having taken grad-level courses is OK if you have a good undergrad math major. And outstanding letters go a long way.

As far as going to a top 25 school is concerned, people get way too obsessed with rankings. There are about fifty R1 programs in the US, and whether you go to the 20th or the 50th matters a lot less than what kind of work you do once you're there. The ranking is also not a reflection at all of the quality of the education you'll get, it only tells you in a very rough sense about the research quality of the faculty and the average talent level of the students. But the important word there is "rough."