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The purpose of investing - at least for me - is to buy quality shares at sensible prices; not to wait for shares to double in a four-year period and then "discover" what a good investment it *would* have been.

This is not a new bull market.
Equities have further downside mean-reversion to do. I doubt the 2009 price lows will fail, but P/E's need to go much lower, otherwise we will have reached "a permanently high plateau" in equity valuations where valuation undershoots no longer occur to balance the overshoots.
FTSE P/E was more than twice the long-run average when it topped in 1999. Now it's a little above the long-run average.

Whatever happened to the mean-reversion principle of equity valuations?
Equal and opposite. For the time and extent the FTSE spent above its long-term average, it must spend an equal time and extent below trend.
If it peaked at twice its long-run average P/E, then it should trough at half its long-run average P/E.
If it spent ten years an average of 20% above trend, it must spend ten years 20% below trend (or 20 years 10% below trend).

Valuations have always mean-reverted before. Historical odds are 100% that they will do so again. Only if "this time is different" will equities not complete the down-cycle of their long-term valuation trends.

It won't be different.
With equities having overshot to the upside by a record extent in 1999-2000 (FTSE100 P/E hit a record around 30x), I'd not at all be surprised if equities eventually undershoot to a near-record extent in the years ahead. Perhaps only beaten by the FT30's P/E ratio of 4x (yield 13%) seen in the mid-1970's.
The last long-running bull-bear cycle lasted from the valuation peak in 1966 through to the valuation trough in 1982. The price lows were halfway along; 1974.
So an on-off valuation mean-reversion process from 2000 to 2016 would be quite reasonable to expect based on past long-term bears - and we already have evidence for price lows half way along, around 2008, just as were seen in the middle of the 1966-82 bear.

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