Is "oftener" obsolete?

Solution 1:

Why yes: Charles Darwin, for one. Or rather, he says oftenest as the superlative. The OED gives this under its entry for often adverb:

Comparative and Superlative.

• 1467 Ordin. Worc. in Eng. Gilds 380 - [They] shullen com and assemble togeder in euery quarter of the yere, ones or oftener and it nede.
• 1558 Bp. Watson Sev. Sacram. xi. 61 - He..that the oftneste and with moste reuerence commeth.
• 1660 Boyle Seraph. Love xvi. (1700) 99 - He is rather welcom’st to God that comes to him oftenest, and stays with him longest.
• A. 1715 Burnet Own Time (1766) I. 246 - As has happened oftner than once before.
• 1784 Cowper Task i. 411 - An idol at whose shrine Who oft’nest sacrifice are favour’d least.
• 1866 Darwin Orig. Spec. (ed. 4) iv. 104 - Those individual flowers..would be oftenest visited by insects, and would be oftenest crossed.

However, with regard to oftentimes, the extended form of ofttimes, the OED notes that the comparative and superlative forms oftener times and oftenest times are now obsolete, last having been seen before the Mayflower cross the Atlantic.

The changeover point was back around the fin de siècle, and not the most recent thereof, either, as shown by this Google N-Gram:

That being said, I myself would use more often and most often as the comparative and superlative degrees of often. In the popularity contest of Google, the two-word versions do come out far in the lead for oftener and even more so with oftenest.

“Out of Sight, Out of Mind”, by Barnabe Googe [1540–1594]

The oftener seen, the more I lust,
The more I lust, the more I smart,
The more I smart, the more I trust,
The more I trust, the heavier heart;
The heavy hearty breeds mine unrest,
Thy absence, therefore, like I best.

The rarer seen, the lest in mind,
The less in mind, the lesser pain,
The lesser pain, less grief I find,
The lesser grief, the greater gain,
The greater gain, the merrier I,
Therefore I wish thy sight to fly.

The further off, the more I joy,
The more I joy, the happier life,
The happier life, less hurts annoy,
The lesser hurts, pleasure most rife:
Such pleasures rife shall I obtain
When distance doth depart us twain.

Solution 2:

When I was thirty-six years old I used the word "oftener" (which was still in very limited usage then) and was promptly interrupted by my five year old daughter with "dad it's 'often' not 'oftener'. I have never used the Apalachian form since and I am now 79