Get the external IP address in shell without dig in 2016?

An alternative is Google DNS:

myip(){   dig @8.8.8.8 -t txt o-o.myaddr.l.google.com |
          grep "client-subnet" |
          grep -o "\([0-9]\{1,3\}\.\)\{3\}\([0-9]\{1,3\}\)" ;   }

If your system does not have dig, then host will be exactly equivalent:

myip(){   host -t txt o-o.myaddr.l.google.com 8.8.8.8 |
          grep -oP "client-subnet \K(\d{1,3}\.){3}\d{1,3}";   }

Both GNU grep (Perl-like -P) and basic (BRE) regex are shown.

And, of course the original site, will also work.
With dig:

myip(){   dig myip.opendns.com @208.67.220.222  |
          grep "^myip\.opendns\.com\."             |
          grep -o "\([0-9]\{1,3\}\.\)\{3\}\([0-9]\{1,3\}\)"  ; }

And with host:

myip(){   host myip.opendns.com 208.67.220.222  |
          grep -oP "^myip\.opendns\.com.* \K(\d{1,3}\.){3}(\d{1,3})" ; }

The two snippets above will work with any of this four OpenDNS resolver addresses:

echo 208.67.22{0,2}.22{0,2}

---

If the direct DNS solution fails in the future, use curl to one of this sites (urls):

IFS=$'\n' read -d '' -a urls <<-'_end_of_text_'
api.ipify.org
bot.whatismyipaddress.com/
canhazip.com/
checkip.dyndns.com/
corz.org/ip
curlmyip.com/
eth0.me/
icanhazip.com/
ident.me/
ifcfg.me/
ifconfig.me/
ip.appspot.com/
ipecho.net/plain
ipof.in/txt
ip.tyk.nu/
l2.io/ip
tnx.nl/ip
wgetip.com/
whatismyip.akamai.com/
_end_of_text_

Call an array address like this:

$ i=5; curl -m10 -L "http://${urls[i]}"
116.132.27.203

In some sites https may also work.