Updated to 30 Sep 2021
It is now common to reach various batting landmarks but miss out on making a century.
Sri Lankan wicketkeeper Niroshan Dickwella now has a record 18 fifties without ever making a century.
Most fifties without a Test century:

The record had stood at 16 by Chetan Chauhan for 40 years.
And one can make 3000 runs without a century (though coming close with a 99):
Most runs without a Test century:

Dickwella is in second place to Shane Warne here. Chauhan is the only other one to cross 2000. Southee still has a chance to reach 2000.
If you look carefully, most of the players in the above list are wicket-keepers or all-rounders. Laird, Brearley and Chauhan are among the exceptions.
We can also look at the
Highest batting averages without a century.
Here we consider with a minimum of 20 innings batted.

Here, the leader is the lesser-known Pakistani batsman Asim Kamal (who made 99 on debut). Current players are TD Paine and N Dickwella (again).
Finally, we look at
Highest score 95-99:

Of these, Mitchell Starc and Dickwella may yet make a century. Other current players who have a chance include AC Agar (not a Test regular), Dilruwan (MDK) Perera and DT Tiripano.
Some scored 95-99 on debut but never made a century. They include:
MDK Perera (primarily a bowler).
VH Stollmeyer (96 in his only Test innings)
Abdul Kadir
AC Agar (At no 11). But not considered a Test regular now.
Asim Kamal (one of only 3 debutants with 99s). The other two were AG Chipperfield and RJ Christiani who also had short Test careers but did make a century apiece.
And finally:
The longest wait for a maiden century:
A Kumble (118th of 132 Tests)
C Vaas (97th of 111 Tests)
SM Pollock (51st of 107 Tests)
ADR Campbell (47th of 60 Tests)