Simple Python username scraper
If you're after speed, I'd suggest scrapy. I was looking for an excuse to try it out and saw your question. When I ran your code on the first 10 pages of the NA leaderboard, it took a little over 4 seconds. Running the below takes about 0.3 seconds, presumably due to initiating all the HTTP requests in parallel:

test.py:

class LolSpider(scrapy.Spider):
    name = 'lolspider'
    start_urls = ['http://lolprofile.net/leaderboards/na/{}'.format(page) for page in range(10)]

    def parse(self, response):
        for name in response.xpath('//a[re:test(@href, "http://lolprofile.net/summoner/")]//text()').extract():
            yield { 'name': name }


Running:

$ scrapy runspider test.py -o names.json


names.json:

[
{"name": "&lt;first name here&gt;"},
{"name": "&lt;second name here&gt;"},
...
]




To actually provide some code review feedback:

import requests # I prefer this and then requests.get over "from requests import get", since "get" is too common a word
from bs4 import BeautifulSoup
import time # ditto here
import re # and here

print('***PYTHON LEAGUE OF LEGENDS USERNAME SCRAPER***')
print('This script scrapes usernames from lolprofile.net')
region = input('Enter the region for scraping(eune/euw/na/br/tr/kr/jp/lan/las/oce/ru)\n')
num_start = int(input('What page to start on? Min 0\n')) # cast to int once here
num_end = int(input('What page to end on? Min starting page + 1\n')) # ditto

size = 0 # use a simple count rather than a list

# Python style dictates snake case
# get the URL rather than set a global variable
def get_url(page_num, region):
    # use string formatting rather than concatenation
    return 'http://lolprofile.net/leaderboards/{}/{}'.format(region, page_num)

def is_ascii(i):
    return all(ord(c) &lt; 128 for c in i)

start = time.time()

# for loop instead of while avoids the need to increment by hand
for page_num in range(num_start, num_end + 1):
    url = get_url(page_num, region)
    print(size)
    html = requests.get(url).text
    soup = BeautifulSoup(html, "html.parser") # html.parser
    # /.* (slash and then anything) rather than /* (any number of slashes) in the regular expression
    user_names = [a.string for a in soup.findAll(href=re.compile('http://lolprofile.net/summoner/.*'))]
    with open('usernames1.txt', 'a') as file:
        for i in user_names:
            if is_ascii(i) and ' ' not in i: # not in
                file.write('%s\n' % i.lower())
                size += 1

end = time.time()

print('{} usernames scraped in a total of {} seconds.'.format(size, end-start))